text:
\section{Introduction}

\noindent The study of higher-spin fields in $2+1$ dimensions has received considerable attention in recent years due to the possibility to overcome some of the difficulties appearing in higher-dimensional cases, where in order to surpass a series of old no-go theorems \cite{Bekaert:2010hw} and obtain a consistent theory, the inclusion of a cosmological constant as well as an infinite tower of interacting higher-spin fields is needed \cite{Vasiliev:1990en,Vasiliev:2003ev}. In fact, in the massless sector, a three-dimensional higher-spin gravity theory with negative cosmological constant including a finite number of higher-spin fields can be defined by means of a Chern-Simons (CS) action \cite{Blencowe:1988gj,Bergshoeff:1989ns,Vasiliev:1995dn}.  Even though CS actions unavoidable lead to no local propagating degrees of freedom in the bulk, higher-spin CS gravity has been shown to admit black hole solutions with non-trivial properties (see the reviews \cite{Bunster:2014mua,Castro:2016tlm} and references therein). Fuerthermore, considering suitable boundary conditions gives rise to rich boundary dynamics and asymptotic symmetries given by $\mathcal W$-algebras \cite{Henneaux:2010xg,Campoleoni:2010zq}. Higher-spin CS theories were later generalized to the case of vanishing cosmological constant \cite{Afshar:2013vka,Gonzalez:2013oaa} and non-Lorentzian gravity \cite{Bergshoeff:2016soe,Chernyavsky:2019hyp}.\\

\noindent The recent interest in higher-spin fields in three space-time dimensions goes beyond gravitational physics. A remarkable $(2+1)$-dimensional system where higher-spin symmetries play an important role is the fractional quantum Hall effect, where $\mathcal W$-algebras appear as the quantum realization of area-preserving diffeormorphism algebra \cite{Cappelli:1992yv,Iso:1992aa}. Moreover, it has been conjectured that, apart from the Magnetoroton \cite{Girvin:1986zz}, a spin-$2$ graviton-like mode, additional collective excitations described by higher-spin fields should be introduced in the known effective field theory descriptions to reproduce the full Girvin-Macdonald-Platzman algebra \cite{Golkar:2016thq,Gromov:2017qeb}. Another exotic scenario presenting higher-spins is Fracton physics (see the reviews \cite{Nandkishore:2018sel,Grosvenor:2021hkn} and references therein), which introduces quasiparticles with restricted mobility whose local interactions are described by higher-rank gauge fields. The higher-spin theories capable of describing these types of physical systems are, however, non-relativistic (NR).\\

\noindent NR symmetries and NR geometry have attracted recent interest due to their applicability in extensions of holography \cite{Taylor:2008tg,Balasubramanian:2008dm,Herzog:2008wg,Bagchi:2009my} and condensed matter physics \cite{Duval:2001hu,Son:2008ye,Geracie:2014nka,Gromov:2014vla}. The extension of kinematical symmetries to include higher-spin generators was introduced in \cite{Bergshoeff:2016soe}, where in particular two different spin-$3$ extensions of the Extended Bargmann and Newton-Hooke algebras were given\footnote{Galilean and Carrollian conformal higher-spin symmetries in arbitrary dimensions have been recently constructed in \cite{Campoleoni:2021blr}.}. On the other hand, Lie algebra expansions methods \cite{Hatsuda:2001pp,deAzcarraga:2002xi,Boulanger:2002bt,Izaurieta:2006zz,deAzcarraga:2007et,Penafiel:2016ufo}  have been extensively used in the construction of extended non-Lorentzian symmetries \cite{Bergshoeff:2019ctr,deAzcarraga:2019mdn,Hansen:2019vqf,Penafiel:2019czp,Gomis:2019nih,Concha:2020eam,Kasikci:2021atn,Gomis:2022spp}. Particularly, in \cite{Penafiel:2019czp} it has been shown that expansion of the Nappi-Witten algebra allows one to define infinite families of NR symmetries in $2+1$ dimensions that generalize the Bargmann and the Newton-Hooke symmetries. These are central extensions of the Generalized Galilean symmetries presented in \cite{Gonzalez:2016xwo} for $D=2+1$. This result can be contrasted with the families of relativistic spin-$3$ symmetries in $2+1$ dimensions found in \cite{Caroca:2017izc} by means of Lie algebra expansions applied to the $\mathfrak{sl}(3,\mathbb R)$ algebra. In particular, two families were contructed explicitly, which define generalized spin-$3$ symmetries of AdS type and Poincar\'e type. In this setup, the first \emph{level} of the expansion yields the spin-$3$ extension of the AdS algebra, $\mathfrak{sl}(3,\mathbb R)\times\mathfrak{sl}(3,\mathbb R)$, and the spin-$3$ extension of the Poincar\'e algebra, respectively. At the second level, spin-$3$ extensions of the $(2+1)$-dimensional version of the Maxwell algebra and the AdS-Lorentz algebra arise, whereas higher levels correspond to spin-$3$ extensions of more complicated examples of the so-called Generalized AdS \cite{Concha:2016kdz} and Generalized Poincar\'e \cite{Izaurieta:2009hz} algebras in three space-time dimensions. Therefore, if higher-spin extensions of the Nappi-Witten symmetry can be defined, expansions of these symmetries can be used to define NR limits of the higher-spin symmetries defined in \cite{Caroca:2017izc}. In $2+1$ dimensions, NR limits usually require to extend relativistic symmetries by introducing extra Abelian generators as a way to obtain a non-degenerate invariant form in the resulting NR algebra (see for example \cite{Bergshoeff:2022eog}). Similarly, to define proper NR contractions of $\mathfrak{sl}(3,\mathbb R)$, it will be necessary to consider suitable extensions thereof. This time, however, the necessary extensions to be introduced are non-Abelian. In particular, we find two higher-spin extension of the $\mathfrak{nw}$ symmetry, which we dub $\mathfrak{hs_3 nw1}$ and $\mathfrak{hs_3 nw2}$, and follow from two inequivalent contractions applied to $\mathfrak{sl}(3,\mathbb R)\oplus \mathfrak{so}(3)$ and $\mathfrak{sl}(3,\mathbb R)\oplus \mathfrak{sl}(2,\mathbb R)\oplus \mathfrak{so}(2)$, respectively.\\
\\
\noindent
In this article, i) we define two different spin-$3$ extensions of the Nappi-Witten symmetry, which we denominate $\mathfrak{hs_3 nw1}$ and $\mathfrak{hs_3 nw2}$, by means of In\"on\"u-Wigner contractions \cite{inonu1993contraction} of suitable direct product extensions of the $\mathfrak{sl}(3,\mathbb R)$ algebra, ii) we show that expansions of these spin-$3$ Nappi-Witten algebras define NR higher-spin symmetries in $2+1$ dimensions, iii) we show that the same result can be achieved by first expanding the $\mathfrak{sl}(3,\mathbb R)$ algebra and subsequently defining a NR contraction of (an extension of) the resulting expanded symmetry, iv) we construct non-degenerate invariant bilinear forms on the resulting NR higher-spin algebras and define the corresponding higher-spin CS gravity theories.\\

\noindent The paper is organized as follows: in section \ref{HSNW} we show how the $\mathfrak{nw}$ algebra can be defined as a NR contraction of the $\mathfrak{sl}(2,\mathbb R)\oplus \mathfrak{so}(2)$ algebra and extend this result to define a NR contraction of the $\mathfrak{sl}(3,\mathbb R)\oplus \mathfrak{so}(3)$ algebra and the $\mathfrak{sl}(3,\mathbb R)\oplus \mathfrak{sl}(2,\mathbb R)\oplus \mathfrak{so}(2)$ algebra, which lead to two different higher-spin extensions of the $\mathfrak{nw}$ symmetry. In section \ref{ExpNW} we construct expanded Nappi-Witten algebras by means of Lie algebra expansions based on Abelian semigroups. The method is also used to define invariant bilinear forms on the expanded algebras and CS actions. In section \ref{Examples}, we show that the spin-$3$ extensions of the Extended Bargmann algebra and the NR Maxwell algebra in $2+1$ dimensions correspond to particular examples of expanded higher-spin Nappi-Witten algebras. By considering a different semigroup, a cosmological constant can be introduced, leading to a spin-$3$ extension of the Newton-Hooke and the NR AdS-Lorentz algebras. The algebraic structure of these symmetries is discussed and the explicit form of the corresponding CS higher-spin gravity theories is given. Finally, we end with a general discussion of our results and elaborate on possible future directions.

\section{Spin-$3$ extensions of the Nappi-Witten algebra}\label{HSNW}

In this section, we construct two different spin-$3$ extensions of the Nappi-Witten algebra $\mathfrak{nw}$ by considering suitable In\"o\"u-Wigner contractions of extensions of the $\mathfrak{sl}(3,\mathbb R)$ algebra. To make our construction more transparent, we start by reviewing how the $\mathfrak{nw}$ algebra itself can be obtained as a contraction of an Abelian extension of $\mathfrak{sl}(2,\mathbb R)$. The starting point is the $\mathfrak{sl}(2,\mathbb R)$ algebra
\begin{equation}
\left[\boldsymbol{L}_A,\boldsymbol{L}_B \right]=\epsilon_{AB}^{\quad C}\boldsymbol{L}_C,\qquad A=0,1,2,
\end{equation}
which is isomorphic to the Lorentz algebra in $2+1$  dimensions, $\mathfrak{so} (2,1)$. Here $\epsilon_{012}=1$ and the relativistic indices are lowered and raised with the Minkowski metric $\eta_{AB}={\rm diag}(-1,1,1)$ and its inverse. A NR limit can be defined by introducing a $\mathfrak{so}(2)$ generator $\boldsymbol{X}$ and redefining the Lie algebra generators in the following way
\begin{equation}\label{slu}
\begin{aligned}
\boldsymbol{L}_{0}  &=  \dfrac{1}{2}\boldsymbol{k}+c^{2}\boldsymbol{\sigma},\qquad
\boldsymbol{L}_a &=  c\,\boldsymbol{k}_a,\qquad
\boldsymbol{X} & =  \frac{1}{2}\boldsymbol{k}-c^{2}\boldsymbol{\sigma} ,
\end{aligned}
\end{equation}
where have introduced the parameter $c$, which stands for the speed of light, and decomposed the relativistic indices as $A=(0,a=1,2)$. The new generators $\left\{\boldsymbol{k}, \boldsymbol{k}_a, \boldsymbol{\sigma}\right\}$ are then expressed in terms of $\boldsymbol{L}_a$ and $\boldsymbol{X}$ by the relations
\begin{equation}\label{invcontNW}
\begin{aligned}
\boldsymbol{k} & =  \boldsymbol{L}_{0}+\boldsymbol{X},\qquad
\boldsymbol{k}_a & =  \frac{1}{c}\boldsymbol{L}_a,\qquad
\boldsymbol{\sigma} & =  \frac{1}{2c^2}\left(\boldsymbol{L}_{0}-\boldsymbol{X}\right),
\end{aligned}
\end{equation}
and, in the limit $c\rightarrow\infty$, their commutators reproduce the Nappi-Witten algebra $\mathfrak{nw}$
\begin{equation}\label{nwalg}
\left[\boldsymbol{k},\boldsymbol{k}_a\right]=\epsilon_{ab}\,\boldsymbol{k}_b
,\qquad
\left[\boldsymbol{k}_a,\boldsymbol{k}_b\right]=-\epsilon_{ab}\,\boldsymbol{\sigma} \qquad (\epsilon_{ab}\equiv\epsilon_{0ab}).
\end{equation}
Due to the central term $\boldsymbol{\sigma}$, the $\mathfrak{nw}$ algebra admits a non-degenerate invariant bilinear form (see Section \ref{IBF}). The Nappi-Witten symmetry is the central extension of the Euclidean symmetry in two dimensions\footnote{This corresponds to the Euclidean version of the Nappi-Witten algebra presented in \cite{Nappi:1993ie}.} \cite{Figueroa-OFarrill:1999cmq}. As we have just shown, it can be interpreted as a centrally extended NR limit of the $\mathfrak{sl}(2,\mathbb R)$ algebra, pretty much in the same way as the Bargmann algebra is a centrally extended NR limit of the Galilean one. It is thus evident that, in order to generalize the previous construction and define a symmetry that could be understood as a NR contraction of $\mathfrak{sl}(3,\mathbb R)$ in the sense of \eqref{slu}, it will be necessary to properly extend the $\mathfrak{sl}(3,\mathbb R)$ algebra. Its commutation relations read\footnote{The convention for the symmetrization of indices is
$x_{(A}y_{B)}=x_A y_B+x_B y_A$, without normalization factor.} \cite{Campoleoni:2010zq}
\begin{equation}\label{sl3alg}
\left[\boldsymbol{L}_A,\boldsymbol{L}_B\right] =\epsilon_{AB}^{\quad C}\boldsymbol{L}_{C}\,\quad
\left[\boldsymbol{L}_A,\boldsymbol{\mathcal L}_{BC}\right] =\epsilon_{A(B}^{\quad \;D}\boldsymbol{\mathcal L}_{C)D}\,\quad
\left[\boldsymbol{\mathcal L}_{AB},\boldsymbol{\mathcal L}_{CD}\right]  =- \eta_{(A(C}\epsilon_{D)B)}^{\quad \;E} \boldsymbol{L}_E,
\end{equation}
where $\boldsymbol{L}_A$ defines an $\mathfrak{sl}(2,\mathbb R)$ sublagelbra and $\boldsymbol{\mathcal L}_{AB}$ is traceless symmetric generator, i.e.
\begin{equation}
\boldsymbol{\mathcal L}_{AB}=\boldsymbol{\mathcal L}_{BA},\quad
\eta^{AB}\boldsymbol{\mathcal L}_{AB}=0\quad\Rightarrow\quad
\boldsymbol{\mathcal L}_{00}=\delta^{ab}\boldsymbol{\mathcal L}_{ab}\equiv \boldsymbol{\mathcal L}_{aa}.
\end{equation}
Splitting the relativistic indices into temporal and spatial components yields
\begin{equation}\label{sltrcomm}
\begin{aligned}
&\left[\boldsymbol{L}_{0},\boldsymbol{L}_a\right]   =\epsilon_{ab}\boldsymbol{L}_b,
&
&\left[\boldsymbol{L}_a,\boldsymbol{L}_b\right]  = -\epsilon_{ab}\boldsymbol{L}_{0},
&
&\left[\boldsymbol{L}_{0},\boldsymbol{\mathcal L}_{a0}\right]   =\epsilon_{ab}\boldsymbol{\mathcal L}_{b0},
\\
&\left[\boldsymbol{L}_{0},\boldsymbol{\mathcal L}_{ab}\right]   =-\epsilon_{c(a}\boldsymbol{\mathcal L}_{b)c},
&
&\left[\boldsymbol{L}_a,\boldsymbol{\mathcal L}_{bc} \right] =-\epsilon_{a(b}\boldsymbol{\mathcal L}_{c)0},
&
&\left[\boldsymbol{L}_a,\boldsymbol{\mathcal L}_{b0}\right]   =-\epsilon_{a(c}\boldsymbol{\mathcal L}_{b)c},
\\
&\left[\boldsymbol{\mathcal L}_{a0},\boldsymbol{\mathcal L}_{b0}\right]  =\epsilon_{ab}\boldsymbol{L}_{0},
&
&\left[\boldsymbol{\mathcal L}_{a0},\boldsymbol{\mathcal L}_{bc}\right]  =\delta_{a(b}\epsilon_{c)d}\boldsymbol{L}_{d},
&
&\left[\boldsymbol{\mathcal L}_{ab},\boldsymbol{\mathcal L}_{cd}\right]  =\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{L}_{0}.
\end{aligned}
\end{equation}
Thus, one would like to keep the definitions \eqref{slu} for the $\mathfrak{sl}(2,\mathbb R)$ generators and find some analog relations for the spin-$3$ generators $\boldsymbol{\mathcal L}_{ab}$ and $\boldsymbol{\mathcal L}_{a0}$ in such a way that the limit  $c\rightarrow$ produces a spin-$3$ extension of $\mathfrak{nw}$ with a non-degenerate invariant form. This requires supplementing the Abelian generator $\boldsymbol{X}$ with extra generators that, as we will see, do not commute with $\boldsymbol{X}$. In the following, we consider two extensions of $\mathfrak{sl}(3,\mathbb R)$ that admit a well-defined NR limit. These are $\mathfrak{sl}(3,\mathbb R)\oplus \mathfrak{so}(3)$ and $\mathfrak{sl}(3,\mathbb R)\oplus \mathfrak{sl}(2,\mathbb R)\oplus \mathfrak{so}(2)$. In each case, NR contractions lead to novel extensions of the $\mathfrak{nw}$ algebra that include higher-rank generators, which can be interpreted has NR higher-spin symmetries.

\subsection{NR contraction of $\mathfrak{sl}(3,\mathbb R)\oplus \mathfrak{so}(3)$}
\label{NRcontractions}

In order to extend the $\mathfrak{sl}(3,\mathbb R)$ commutation relations \eqref{sltrcomm}, we introduce a $\mathfrak{so}(3)$ algebra spanned by generators $\{\boldsymbol{X},\boldsymbol{X}_a\}$ satisfying
\begin{equation}\label{so3alg}
\begin{aligned}
\left[\boldsymbol{X},\boldsymbol{X}_a\right]=&\epsilon_{ab}\boldsymbol{X}_b
,\qquad
\left[\boldsymbol{X}_a, \boldsymbol{X}_b\right]=&\epsilon_{ab} \boldsymbol{X},
\end{aligned}
\end{equation}
and consider the direct product $\mathfrak{sl}(3,\mathbb{R})\times\mathfrak{so}(3)$ generated by $\boldsymbol{L}_A=\{\boldsymbol{L}_0,\boldsymbol{L}_a\}$, $\boldsymbol{\mathcal L}_{AB}= \{ \boldsymbol{\mathcal L}_{a0},\boldsymbol{\mathcal L}_{ab}\}$, and $\{\boldsymbol{X}, \boldsymbol{X}_a\}$. The NR limit can be defined by using the change of basis
\begin{equation}\label{redefreli}
\begin{aligned}
\boldsymbol{L}_{0}&= \dfrac{1}{2}\boldsymbol{k}+c^{2}\boldsymbol{\sigma},
&
\boldsymbol{L}_a&=c\, \boldsymbol{k}_a,
&
\boldsymbol{X} &=\dfrac{1}{2}\boldsymbol{k}-c^{2}\boldsymbol{\sigma},
\\
\boldsymbol{\mathcal L}_{a0} &=\dfrac{1}{2}\boldsymbol{\mathcal K}_a+c^{2}\boldsymbol{\Sigma}_a,
&
\boldsymbol{\mathcal L}_{ab} &= c\,\boldsymbol{\mathcal K}_{ab},
&
\boldsymbol{X}_a&=\dfrac{1}{2}\boldsymbol{\mathcal K}_a-c^{2}\boldsymbol{\Sigma}_a,
\end{aligned}
\end{equation}
with inverse relations given by
\begin{equation}\label{gencont1}
\begin{aligned}
\boldsymbol{k} & =  \boldsymbol{L}_{0}+\boldsymbol{X}, &
\boldsymbol{k}_a  & =  \frac{1}{c} \boldsymbol{L}_a, &
\boldsymbol{\sigma} & =  \frac{1}{2c^2}\left(\boldsymbol{L}_{0}-\boldsymbol{X}\right),
\\
\boldsymbol{\mathcal K}_a&= \boldsymbol{\mathcal L}_{a0}+\boldsymbol{X}_a,
&
\boldsymbol{\mathcal K}_{ab}&=  \frac{1}{c}\boldsymbol{\mathcal L}_{ab},
&
\boldsymbol{\Sigma}_a&=  \frac{1}{2c^{2}}\left(\boldsymbol{\mathcal L}_{a0}-\boldsymbol{X}_a\right).
\end{aligned}
\end{equation}
In the limit $c\rightarrow\infty$ we find the following non-vanishing commutation relations for the generators \eqref{gencont1}
\begin{equation}\label{HSNWalgebra1}
\begin{aligned}
&\left[\boldsymbol{k}, \boldsymbol{k}_a\right]  =  \epsilon_{ab} \boldsymbol{k}_b,&
&\left[\boldsymbol{k}_a,\boldsymbol{k}_b\right] = -\epsilon_{ab}\boldsymbol{\sigma},&
&\left[\boldsymbol{k},\boldsymbol{\mathcal K}_a\right]  = \epsilon_{ab}\boldsymbol{\mathcal K}_b,
\\
&\left[\boldsymbol{k},\boldsymbol{\mathcal K}_{ab}\right] =  -\epsilon_{c(a}\boldsymbol{\mathcal K}_{b)c},&
&\left[\boldsymbol{k},\boldsymbol{\Sigma}_a\right] = \epsilon_{ab}\boldsymbol{\Sigma}_b,&
&\left[\boldsymbol{k}_a,\boldsymbol{\mathcal K}_b\right] = -\epsilon_{a(c}\boldsymbol{\mathcal K}_{b)c},
\\
&\left[\boldsymbol{k}_a,\boldsymbol{\mathcal K}_{bc}\right]  =  -\epsilon_{a(b}\boldsymbol{\Sigma}_{c)}, &
&\left[\boldsymbol{\sigma},\boldsymbol{\mathcal K}_a\right]  =  \epsilon_{ab}\boldsymbol{\Sigma}_b,&
&\left[\boldsymbol{\mathcal K}_a,\boldsymbol{\mathcal K}_b\right]  =  \epsilon_{ab}\boldsymbol{k},
\\
&\left[\boldsymbol{\mathcal K}_a, \boldsymbol{\Sigma}_b\right] = \epsilon_{ab}\boldsymbol{\sigma},&
&\left[\boldsymbol{\mathcal K}_a,\boldsymbol{\mathcal K}_{bc}\right]  =\delta_{a(b}\epsilon_{c)d}\boldsymbol{k}_d,&
&\left[\boldsymbol{\mathcal K}_{ab},\boldsymbol{\mathcal K}_{cd}\right] = \delta_{(a(c}\epsilon_{d)b)}\boldsymbol{\sigma}.
\end{aligned}
\end{equation}
We will refer to this algebra as $\mathfrak{hs_3 nw1}$ since it defines a higher-spin extension of the Nappi-Witten algebra\footnote{Note that $\mathfrak{hs_3 nw1}$ is isomorphic to the homogeneous part of spin-$3$ Bargmann algebra in $2+1$ dimensions $\mathfrak{hs_3 barg1_{2+1}}$ found in \cite{Bergshoeff:2016soe}.} .

\subsection{NR contraction of $\mathfrak{sl}(3,\mathbb R)\oplus \mathfrak{sl}(2,\mathbb R)\oplus \mathfrak{so}(2)$}
As a second possibility to extend $\mathfrak{sl}(3,\mathbb R)$, we introduce the $\mathfrak{sl}(2,\mathbb R)\oplus \mathfrak{so}(2)$ algebra
\begin{equation}\label{sl2xu1v1}
[\boldsymbol{\mathcal A},\boldsymbol{\mathcal B}]=\boldsymbol{\mathcal B},\quad [\boldsymbol{\mathcal A},\boldsymbol{\mathcal C}]=-\boldsymbol{\mathcal C},\quad [\boldsymbol{\mathcal B},\boldsymbol{\mathcal C}]=2\boldsymbol{\mathcal A},\quad [\boldsymbol{\mathcal D},{\rm all}]=0,
\end{equation}
where the set $\{\boldsymbol{\mathcal A},\boldsymbol{\mathcal B},\boldsymbol{\mathcal C}\}$ spans $\mathfrak{sl}(2,\mathbb R)$, whereas $\boldsymbol{\mathcal D}$ is the Abelian $\mathfrak{so}(2)$ generator. By redefining the generators as
\begin{equation}
\begin{aligned}
\boldsymbol{\mathcal A}&=\frac{1}{4}\left( \boldsymbol{X}_{22} -\boldsymbol{X}_{11}\right),&
\boldsymbol{\mathcal B}&=\frac{1}{2}\left( \boldsymbol{X}_{12} +\boldsymbol{X}\right),\\
\boldsymbol{\mathcal C}&=\frac{1}{2}\left( \boldsymbol{X}_{12} -\boldsymbol{X}\right),&
\boldsymbol{\mathcal D}&=\frac{1}{4}\left( \boldsymbol{X}_{22} +\boldsymbol{X}_{11}\right),\\
\end{aligned}
\end{equation}
the $\mathfrak{sl}(2,\mathbb R)\oplus \mathfrak{so}(2)$ algebra \eqref{sl2xu1v1} can be written in a form that is more convenient for our pourposes, namely
\begin{equation}\label{sl2xu1v2}
\begin{aligned}
\left[\boldsymbol{X},\boldsymbol{X}_{ab}\right]=&-\epsilon_{c(a}\boldsymbol{X}_{b)c}
,\qquad
\left[\boldsymbol{X}_{ab},\boldsymbol{X}_{cd}\right]=&\delta_{(a(c}\epsilon_{d)b)} \boldsymbol{X},
\end{aligned}
\end{equation}
with $\boldsymbol{X}_{ab}=\boldsymbol{X}_{ba}$. Thus, we consider the $\mathfrak{sl}(3,\mathbb R)\oplus \mathfrak{sl}(2,\mathbb R)\oplus \mathfrak{so}(2)$ algebra generated by $\boldsymbol{L}_A=\{\boldsymbol{L}_0,\boldsymbol{L}_a\}$, $\boldsymbol{\mathcal L}_{AB}= \{ \boldsymbol{\mathcal L}_{a0},\boldsymbol{\mathcal L}_{ab}\}$, which satisfy the commutation relations \eqref{sltrcomm}, and $\{\boldsymbol{X}, \boldsymbol{X}_{ab}\}$. The NR contraction follows from the generator redefinition
\begin{equation}\label{redefrelii}
\begin{aligned}
\boldsymbol{L}_{0}&= \dfrac{1}{2}\boldsymbol{k}+c^{2}\boldsymbol{\sigma},
&
\boldsymbol{L}_a &=c\,\boldsymbol{k}_a ,
&
\boldsymbol{X} &=\dfrac{1}{2}\boldsymbol{k}-c^{2}\boldsymbol{\sigma},
\\
\boldsymbol{\mathcal L}_{ab} &=\dfrac{1}{2}\tilde{\boldsymbol{\mathcal K}}_{ab}+c^{2}\tilde{\boldsymbol{\Sigma}}_{ab},
&
\boldsymbol{\mathcal L}_{a0} &= c\,\tilde{\boldsymbol{\mathcal K}}_a ,
&
\boldsymbol{X}_{ab}&=\dfrac{1}{2}\tilde{\boldsymbol{\mathcal K}}_{ab}-c^{2}\tilde{\boldsymbol{\Sigma}}_{ab}.
\end{aligned}
\end{equation}
analog to \eqref{redefreli}. Inverting the transformation yields the relations
\begin{equation}\label{gencont2}
\begin{aligned}
\boldsymbol{k} & =  \boldsymbol{L}_{0}+\boldsymbol{X}, &
\boldsymbol{k}_a & =  \frac{1}{c}\boldsymbol{L}_a, &
\boldsymbol{\sigma} & =  \frac{1}{2c^2}\left(\boldsymbol{L}_{0}-\boldsymbol{X}\right),
\\
\tilde{\boldsymbol{\mathcal K}}_{ab}&= \boldsymbol{\mathcal L}_{ab}+\boldsymbol{X}_{ab},
&
\tilde{\boldsymbol{\mathcal K}}_a &=  \frac{1}{c}\boldsymbol{\mathcal L}_{a0},
&
\tilde{\boldsymbol{\Sigma}}_{ab}&=  \frac{1}{2c^{2}}\left(\boldsymbol{\mathcal L}_{ab}-\boldsymbol{X}_{ab}\right).
\end{aligned}
\end{equation}
which can be used to find the commutation relations in the new basis. In the limit $c\rightarrow\infty$ the non-vanishing commutators read
\begin{equation}\label{HSNWalgebra2}
\begin{aligned}
&[\boldsymbol{k},\boldsymbol{k}_a ]    =\epsilon_{ab}\boldsymbol{k}_b ,&
&[\boldsymbol{k}_a ,\boldsymbol{k}_b ] = -\epsilon_{ab}\boldsymbol{\sigma},&
&[\boldsymbol{k},\tilde{\boldsymbol{\mathcal K}}_a ] =\epsilon_{ab}\tilde{\boldsymbol{\mathcal K}}_b ,
\\
&[\boldsymbol{k},\tilde{\boldsymbol{\mathcal K}}_{ab}] =-\epsilon_{c(a}\tilde{\boldsymbol{\mathcal K}}_{b)c},&
&[\boldsymbol{k},\tilde{\boldsymbol{\Sigma}}_{ab}] =-\epsilon_{c(a}\tilde{\boldsymbol{\Sigma}}_{b)c},&
&[\boldsymbol{k}_a ,\tilde{\boldsymbol{\mathcal K}}_b ]=-\epsilon_{a(c}\tilde{\boldsymbol{\Sigma}}_{b)c},
\\
&[\boldsymbol{k}_a ,\tilde{\boldsymbol{\mathcal K}}_{bc}]=-\epsilon_{a(b}\tilde{\boldsymbol{\mathcal K}}_{c)},&
&[\boldsymbol{\sigma},\tilde{\boldsymbol{\mathcal K}}_{ab}] =-\epsilon_{c(a}\tilde{\boldsymbol{\Sigma}}_{b)c},&
&[\tilde{\boldsymbol{\mathcal K}}_{ab},\tilde{\boldsymbol{\mathcal K}}_{cd}] =\delta_{(a(c}\epsilon_{d)b)} \boldsymbol{k}.
\\
&[\tilde{\boldsymbol{\mathcal K}}_a ,\tilde{\boldsymbol{\mathcal K}}_b ] =\epsilon_{ab}\boldsymbol{\sigma},&
&[\tilde{\boldsymbol{\mathcal K}}_a ,\tilde{\boldsymbol{\mathcal K}}_{bc}] =\delta_{a(b}\epsilon_{c)d}\boldsymbol{k}_d,&
&[\tilde{\boldsymbol{\mathcal K}}_{ab},\tilde{\boldsymbol{\Sigma}}_{cd}] =\delta_{(a(c}\epsilon_{d)b)} \boldsymbol{\sigma},
\end{aligned}
\end{equation}
and define a second higher-spin extension of the Nappi-Witten symmetry\footnote{As in the previous case, the algebra $\mathfrak{hs_3 nw2}$ is isomorphic to the homogenous part of the second spin-$3$ extension of the Extended Bargmann algebra in $2+1$ dimensions $\mathfrak{hs_3 barg2_{2+1}}$ found in \cite{Bergshoeff:2016soe}, which can be shown using the redefinition given in Eq. \eqref{redefSigma}.}, which we dub as $\mathfrak{hs_3 nw2}$. Note that is is possible to define a new generator $\tilde{\boldsymbol{\Sigma}}_{ab}$
\begin{equation}\label{redefSigma}
\tilde{\boldsymbol{\Sigma}}_{ab}\;\longrightarrow\;\tilde{\boldsymbol{\Sigma}}^{\prime}_{ab}=\tilde{\boldsymbol{\Sigma}}_{ab} +\delta_{ab} \tilde{\boldsymbol{\Sigma}}_{cc} ,
\end{equation}
which has the effect of modifying one of the commutation relations in \eqref{HSNWalgebra2} as
\begin{equation}
[\boldsymbol{k}_a ,\tilde{\boldsymbol{\mathcal K}}_b ]=-\epsilon_{a(c}\tilde{\boldsymbol{\Sigma}}_{b)c}
\quad\longrightarrow\quad
[\boldsymbol{k}_a ,\tilde{\boldsymbol{\mathcal K}}_b ]=-\epsilon_{ac}\tilde{\boldsymbol{\Sigma}}^{\prime}_{bc},
\end{equation}
while leaving the other commutation relations involving $\tilde{\boldsymbol{\Sigma}}_{ab}$ invariant. This relation will be useful in Section \ref{Examples}, when connecting some of our results with the ones found in \cite{Bergshoeff:2016soe}.

\section{Three-dimensional higher-spin gravity from spin-$3$ Nappi-Witten algebras}
\label{ExpNW}

In the previous section we obtained two different higher-spin extensions of the $\mathfrak{nw}$ algebra, namely $\mathfrak{hs_3 nw1}$ and $\mathfrak{hs_3 nw2}$, by defining NR contractions of suitable extensions of $\mathfrak{sl}(3,\mathbb R)$. This can be summarized as follows:
\begin{equation}
\begin{aligned}
&\mathfrak{sl}(3,\mathbb R)\oplus \mathfrak{so}(3)\;&
\underset{c\rightarrow\infty}{\longrightarrow}& \qquad\mathfrak{hs_3 nw1}
\\[6pt]
&\mathfrak{sl}(3,\mathbb R)\oplus \mathfrak{sl}(2,\mathbb R)\oplus \mathfrak{so}(2)\;&\underset{c\rightarrow\infty}{\longrightarrow}& \qquad\mathfrak{hs_3 nw2}
\end{aligned}
\end{equation}
The next step in our construction is to apply the Lie algebra expansion method based on Abelian semigroups \cite{Izaurieta:2006zz} to these NR symmetries and construct CS actions invariant under them.

\subsection{Expansions of higher-spin Nappi-Witten algebras}
In order to define expansions of the spin-$3$ Nappi-Witten algebras \eqref{HSNWalgebra1} and \eqref{HSNWalgebra2}, we introduce an Abelian semigroup $S=\left\{ \lambda _{0},\lambda _{1},\dots
,\lambda _{N}\right\} $, endowed with the associative product \begin{equation}\label{slaw}
\lambda_i \cdot \lambda_j= \lambda_j \cdot \lambda_i = \lambda_{i\diamond j}\,,
\end{equation}
where $\diamond$ defines the semigroup product law. Since the direct product of $S$ with a Lie algebra is again a Lie algebra \cite{Izaurieta:2006zz}, one can define the expanded Nappi-Witten algebra \cite{Penafiel:2019czp}
\begin{equation}\label{expnw}
S\times \mathfrak{nw}=\mathrm{Span}\left\{ S\otimes \{
\boldsymbol{k},\boldsymbol{k}_a,\boldsymbol{\sigma} \} \right\}.
\end{equation}
whose generators are obtained by considering the tensor product of the semigroup elements and the generators of the $\mathfrak{nw}$ algebra, i.e.
\begin{equation}\label{expandedgen}
\boldsymbol{k}^{(i)}_a=\lambda_i \otimes\boldsymbol{k}_{a}\,,\quad
\boldsymbol{k}^{(i)}=\lambda_i\otimes\boldsymbol{k} \,,\quad
\boldsymbol{\sigma}^{(i)} =\lambda_i \otimes\boldsymbol{\sigma}\,,
\end{equation}
The commutation relations of the expanded Nappi-Witten algebra are
\begin{equation}\label{expNW}
[ \boldsymbol{k}^{(i)},\boldsymbol{k}^{(j)}_a ] = \epsilon_{a}^{\;\;b} \boldsymbol{k}^{(i\diamond j)}_b , \quad
[ \boldsymbol{k}^{(i)}_a,\boldsymbol{k}^{(j)}_b ] = -\epsilon_{ab}\boldsymbol{\sigma}^{(i\diamond j)} .
\end{equation}
Similarly, one can apply the expansion procedure to the symmetries $\mathfrak{hs_3 nw1}$ and $\mathfrak{hs_3 nw2}$, which were defined in Eqs. \eqref{HSNWalgebra1} and \eqref{HSNWalgebra2}, as follows:

\begin{itemize}
\item \textbf{Expansion of $\mathfrak{hs_3 nw1}$}: we consider the direct product of $S$ and the generators of \eqref{HSNWalgebra1}, which has the form
\begin{equation}\label{exp1}
S\times \mathfrak{hs_3 nw1}=\mathrm{Span}\left\{ S\otimes \{
\boldsymbol{k},\boldsymbol{k}_a,\boldsymbol{\sigma} ,\boldsymbol{\mathcal K} _a,\boldsymbol{\mathcal K} _{ab},\boldsymbol{\Sigma} _a \} \right\},
\end{equation}
with the expanded generators defined in \eqref{expandedgen} plus
\begin{equation}\label{expandedgenhs1}
\boldsymbol{\mathcal K}^{(i)}_{ab}=\lambda_i \otimes\boldsymbol{\mathcal K}_{ab}\,,\quad
\boldsymbol{\mathcal K}_a^{(i)}=\lambda_i\otimes\boldsymbol{\mathcal K}_a \,,\quad
\boldsymbol{\Sigma}_a^{(i)} =\lambda_i \otimes\boldsymbol{\Sigma}_a\,.
\end{equation}
The commutation relations of the $S\times \mathfrak{hs_3 nw1}$ algebra are directly obtained from \eqref{HSNWalgebra1} and the semigroup multiplication law \eqref{slaw}, which give
\begin{equation}\label{HSNWalgebra1exp}
\begin{aligned}
&[\boldsymbol{k}^{(i)}, \boldsymbol{k}_a^{(j)}]  =  \epsilon_{ab} \boldsymbol{k}_b^{(i\diamond j)},&
&[\boldsymbol{k}_a^{(i)},\boldsymbol{k}_b^{(j)}] = -\epsilon_{ab}\boldsymbol{\sigma}^{(i\diamond j)},&
&[\boldsymbol{k}^{(i)},\boldsymbol{\mathcal K}_a^{(j)}]  = \epsilon_{ab}\boldsymbol{\mathcal K}_b^{(i\diamond j)},
\\
&[\boldsymbol{k}^{(i)},\boldsymbol{\mathcal K}_{ab}^{(j)}] =  -\epsilon_{c(a}\boldsymbol{\mathcal K}_{b)c}^{(i\diamond j)},&
&[\boldsymbol{k}^{(i)},\boldsymbol{\Sigma}_a^{(j)}] = \epsilon_{ab}\boldsymbol{\Sigma}_b^{(i\diamond j)},&
&[\boldsymbol{k}_a^{(i)},\boldsymbol{\mathcal K}_b^{(j)}] = -\epsilon_{a(c}\boldsymbol{\mathcal K}_{b)c}^{(i\diamond j)},
\\
&[\boldsymbol{k}_a^{(i)},\boldsymbol{\mathcal K}_{bc}^{(j)}]  =  -\epsilon_{a(b}\boldsymbol{\Sigma}_{c)}^{(i\diamond j)}, &
&[\boldsymbol{\sigma}^{(i)},\boldsymbol{\mathcal K}_a^{(j)}]  =  \epsilon_{ab}\boldsymbol{\Sigma}_b^{(i\diamond j)},&
&[\boldsymbol{\mathcal K}_a^{(i)},\boldsymbol{\mathcal K}_b^{(j)}]  =  \epsilon_{ab}\boldsymbol{k}^{(i\diamond j)},
\\
&[\boldsymbol{\mathcal K}_a^{(i)}, \boldsymbol{\Sigma}_b^{(j)}] = \epsilon_{ab}\boldsymbol{\sigma}^{(i\diamond j)},&
&[\boldsymbol{\mathcal K}_a^{(i)},\boldsymbol{\mathcal K}_{bc}^{(j)}]  =\delta_{a(b}\epsilon_{c)d}\boldsymbol{k}_d^{(i\diamond j)},&
&[\boldsymbol{\mathcal K}_{ab}^{(i)},\boldsymbol{\mathcal K}_{cd}^{(j)}] = \delta_{(a(c}\epsilon_{d)b)}\boldsymbol{\sigma}^{(i\diamond j)}.
\end{aligned}
\end{equation}

\item \textbf{Expansion of $\mathfrak{hs_3 nw2}$}: applying the same method to \eqref{HSNWalgebra2} leads to
\begin{equation}\label{exp2}
S\times \mathfrak{hs_3 nw2}=\mathrm{Span}\left\{ S\otimes  \{
\boldsymbol{k},\boldsymbol{k}_a,\boldsymbol{\sigma} ,\tilde{\boldsymbol{\mathcal K}} _a, \tilde{\boldsymbol{\mathcal K}} _{ab}, \tilde{\boldsymbol{\Sigma}} _{ab} \} \right\}.
\end{equation}
The expanded generators, in this case, are given by \eqref{expandedgen} together with
\begin{equation}\label{expandedgenhs2}
\tilde{\boldsymbol{\mathcal K}}^{(i)}_{ab}=\lambda_i \otimes\tilde{\boldsymbol{\mathcal K}}_{ab}\,,\quad
\tilde{\boldsymbol{\mathcal K}}_a^{(i)}=\lambda_i\otimes\tilde{\boldsymbol{\mathcal K}}_a \,,\quad
\tilde{\boldsymbol{\Sigma}}_{ab}^{(i)} =\lambda_i \otimes\tilde{\boldsymbol{\Sigma}}_{ab}\,,
\end{equation}
and the commutation relations of $S\times \mathfrak{hs_3 nw2}$ read
\begin{equation}\label{HSNWalgebra2exp}
\begin{aligned}
&[\boldsymbol{k}^{(i)},\boldsymbol{k}_a^{(j)} ]    =\epsilon_{ab}\boldsymbol{k}_b^{(i\diamond j)},&
&[\boldsymbol{k}_a^{(i)} ,\boldsymbol{k}_b^{(j)} ] = -\epsilon_{ab}\boldsymbol{\sigma}^{(i\diamond j)},&
&[\boldsymbol{k}^{(i)},\tilde{\boldsymbol{\mathcal K}}_a^{(j)} ] =\epsilon_{ab}\tilde{\boldsymbol{\mathcal K}}_b^{(i\diamond j)} ,
\\
&[\boldsymbol{k}^{(i)},\tilde{\boldsymbol{\mathcal K}}_{ab}^{(j)}] =-\epsilon_{c(a}\tilde{\boldsymbol{\mathcal K}}_{b)c}^{(i\diamond j)},&
&[\boldsymbol{k}^{(i)},\tilde{\boldsymbol{\Sigma}}_{ab}^{(j)}] =-\epsilon_{c(a}\tilde{\boldsymbol{\Sigma}}_{b)c}^{(i\diamond j)},&
&[\boldsymbol{k}_a^{(i)} ,\tilde{\boldsymbol{\mathcal K}}_b^{(j)} ]=-\epsilon_{a(c}\tilde{\boldsymbol{\Sigma}}_{b)c}^{(i\diamond j)},
\\
&[\boldsymbol{k}_a^{(i)} ,\tilde{\boldsymbol{\mathcal K}}_{bc}^{(j)}]=-\epsilon_{a(b}\tilde{\boldsymbol{\mathcal K}}_{c)}^{(i\diamond j)},&
&[\boldsymbol{\sigma}^{(i)},\tilde{\boldsymbol{\mathcal K}}_{ab}^{(j)}] =-\epsilon_{c(a}\tilde{\boldsymbol{\Sigma}}_{b)c}^{(i\diamond j)},&
&[\tilde{\boldsymbol{\mathcal K}}_{ab}^{(i)},\tilde{\boldsymbol{\mathcal K}}_{cd}^{(j)}] =\delta_{(a(c}\epsilon_{d)b)} \boldsymbol{k}^{(i\diamond j)},
\\
&[\tilde{\boldsymbol{\mathcal K}}_a^{(i)} ,\tilde{\boldsymbol{\mathcal K}}_b^{(j)} ] =\epsilon_{ab}\boldsymbol{\sigma}^{(i\diamond j)},&
&[\tilde{\boldsymbol{\mathcal K}}_a^{(i)} ,\tilde{\boldsymbol{\mathcal K}}_{bc}^{(j)}] =\delta_{a(b}\epsilon_{c)d}\boldsymbol{k}_d^{(i\diamond j)},&
&[\tilde{\boldsymbol{\mathcal K}}_{ab}^{(i)},\tilde{\boldsymbol{\Sigma}}_{cd}^{(j)}] =\delta_{(a(c}\epsilon_{d)b)} \boldsymbol{\sigma}^{(i\diamond j)}.
\end{aligned}
\end{equation}
We should note that, as done in \eqref{redefSigma} in the case of the $\mathfrak{hs_3 nw2}$ algebra, one can redefine the expanded generator
\begin{equation}\label{redefSigmaexp}
\tilde{\boldsymbol{\Sigma}}_{ab}^{(i)}\;\longrightarrow\;\tilde{\boldsymbol{\Sigma}}^{\prime(i)}_{ab}=\tilde{\boldsymbol{\Sigma}}_{ab}^{(i)} +\delta_{ab} \tilde{\boldsymbol{\Sigma}}_{cc}^{(i)},
\end{equation}
and modify the commutator of the generators $\boldsymbol{k}_a^{(i)}$ and $\tilde{\boldsymbol{\mathcal K}}_a^{(i)} $ in \eqref{HSNWalgebra2exp} as
\begin{equation}
[\boldsymbol{k}_a^{(i)} ,\tilde{\boldsymbol{\mathcal K}}_b^{(j)} ]=-\epsilon_{a(c}\tilde{\boldsymbol{\Sigma}}_{b)c}^{(i\diamond j}
\quad\longrightarrow\quad
[\boldsymbol{k}_a^{(i)} ,\tilde{\boldsymbol{\mathcal K}}_b^{(j)} ]=-\epsilon_{ac}\tilde{\boldsymbol{\Sigma}}^{\prime(i\diamond j)}_{bc},
\end{equation}
while leaving all the remaining commutation relations of $S\times\mathfrak{hs_3 nw2}$ invariant.

\end{itemize}
The algebras defined by the commutation relations \eqref{HSNWalgebra1exp} and \eqref{HSNWalgebra2exp} define two different higher-spin extensions of the expanded Nappi-Witten algebra \eqref{expNW}. We know from Ref. \cite{Penafiel:2019czp} that different choices of the semigroup $S$ lead to the expanded Nappi-Witten algebras $S\times \mathfrak{nw}$ isomorphic to NR symmetries in $2+1$ dimensions such as the Extended Bargmann \cite{Papageorgiou:2009zc}, Exotic Newton-Hooke \cite{Alvarez:2007fw} and NR Maxwell \cite{Aviles:2018jzw} algebras. Higher-spin extension of these NR symmetries can be obtained by considering the same semigroups in the expanded algebras \eqref{HSNWalgebra1exp} and \eqref{HSNWalgebra2exp}.

\subsection{Invariant bilinear forms}
\label{IBF}

The In\"on\"u-Wigner contractions introduced in Section \ref{HSNW} can be used to define non-degenerate invariant bilinear forms on the spin-$3$ Nappi-Witten algebras $\mathfrak{hs_3 nw1}$ and $\mathfrak{hs_3 nw2}$. We begin by showing how this procedure works in the case of the Nappi-Witten algebra \eqref{nwalg}, by looking again at the contraction \eqref{slu} of the $\mathfrak{sl}(2,\mathbb R)\oplus \mathfrak{so}(2)$ algebra, whose invariant form is given by
\begin{equation}\label{itsltwu1}
\left \langle \boldsymbol{L}_A , \boldsymbol{L}_B \right \rangle =\varepsilon_0\eta _{AB},\qquad
\left \langle \boldsymbol{X}, \boldsymbol{X} \right \rangle =\varepsilon_1,
\end{equation}
where $\varepsilon_0$ and  $\varepsilon_1$ are non-vanishing constants. Introducing the constants
\begin{equation}\label{redefitsltwu1}
\gamma_0= \frac{1}{c^2}\varepsilon_0,\qquad \gamma_1= \varepsilon_1-\varepsilon_0,
\end{equation}
the generator redefinition \eqref{invcontNW} yields the invariant bilinear form on the $\mathfrak{nw}$ algebra \cite{Figueroa-OFarrill:1999cmq} in the
limit $c\rightarrow\infty$, which is non-degenerate for $\gamma_0\neq0$:
\begin{equation}\label{itnw}
\left \langle \boldsymbol{k},\boldsymbol{\sigma} \right \rangle = - \gamma_0 ,\quad \left \langle \boldsymbol{k}_a,
\boldsymbol{k}_b \right \rangle = \gamma_0\delta_{ab} ,\quad \left \langle \boldsymbol{k},\boldsymbol{k} \right
\rangle = \gamma_1.
\end{equation}

To define invariant tensors on $\mathfrak{hs_3 nw1}$ and $\mathfrak{hs_3 nw2}$, we follow exactly the same procedure. The non-degenerate invariant bilinear form of the $\mathfrak{sl}(3,\mathbb R)$ is \cite{Campoleoni:2010zq}
\begin{equation}\label{itsltr}
\left \langle \boldsymbol{L}_A , \boldsymbol{L}_B\right \rangle =\varepsilon_0\eta _{AB},\qquad
\left \langle \boldsymbol{\mathcal L}_{AB} \boldsymbol{\mathcal L}_{CD}\right \rangle
=\varepsilon_0 \left( \eta _{A(C}\eta _{D)B}-\frac{2}{3}\eta
_{AB}\eta _{CD} \right),
\end{equation}
which, after splitting temporal and spatial components, takes the form
\begin{equation}\label{invtsl3}
\left\langle \boldsymbol{L}_{0}\boldsymbol{L}_{0}\right \rangle
=-\varepsilon_0,\quad
\left\langle \boldsymbol{L}_{a}\boldsymbol{L}_{b}\right \rangle=-
\left\langle \boldsymbol{\mathcal L}_{a0}\boldsymbol{\mathcal L}_{b0}\right \rangle
=\varepsilon_0 \delta _{ab}
,\quad
\left \langle
\boldsymbol{\mathcal L}_{ab}\boldsymbol{\mathcal L}_{cd}\right \rangle =\varepsilon_0 \left( \delta _{a(c}\delta _{d)b}-\frac{2}{3}\delta _{ab}\delta _{cd} \right).
\end{equation}
The extensions we are interested in require the introduction of invariant tensors on $\mathfrak{so}(3)$ and $\mathfrak{sl}(2,\mathbb R)\oplus \mathfrak{so}(2)$ before taking the NR limit.
\begin{itemize}
\item The invariant bilinear form on $\mathfrak{so}(3)$ can be written as
\begin{equation}
\left \langle \boldsymbol{X},\boldsymbol{X}\right \rangle =\varepsilon_1, \qquad \left \langle
\boldsymbol{X}_a,\boldsymbol{X}_b\right \rangle =\varepsilon_1 \delta _{ab}
\end{equation}
Putting this together with \eqref{invtsl3} defines the invariant form on $\mathfrak{sl}(3,\mathbb R)\oplus \mathfrak{so}(3)$. Using \eqref{redefitsltwu1}, \eqref{gencont1} and taking the limit $c\rightarrow\infty$, one finds that the invariant tensor for the generators of the $\mathfrak{hs_3 nw1}$ algebra \eqref{HSNWalgebra1} is given by \eqref{itnw} plus
\begin{equation}\label{invtcont1}
\left\langle \boldsymbol{\mathcal K}_a  , \boldsymbol{\Sigma}_b \right\rangle= -\gamma_0 \delta_{ab} ,\quad
\left\langle \boldsymbol{\mathcal K}_{ab} ,  \boldsymbol{\mathcal K}_{cd} \right\rangle= \gamma_0 \left( \delta _{a(c}\delta _{d)b}-\frac{2}{3}\delta _{ab}\delta _{cd} \right)  ,\quad
\left\langle \boldsymbol{\mathcal K}_a  ,\boldsymbol{\mathcal K}_b \right\rangle= \gamma_1  \delta_{ab},
\end{equation}
which is non-degenerate provided $\gamma_0\neq0$.

\item The invariant bilinear form on the $\mathfrak{sl}(2,\mathbb R)\oplus \mathfrak{so}(2)$ algebra \eqref{sl2xu1v1} is given by
\begin{equation}
\left\langle \boldsymbol{\mathcal A},\boldsymbol{\mathcal A}\right\rangle=\frac{1}{4}\sigma_0,
\qquad
\left\langle \boldsymbol{\mathcal B},\boldsymbol{\mathcal C}\right\rangle= \frac{1}{2}\sigma_0,
\qquad
\left\langle \boldsymbol{\mathcal D},\boldsymbol{\mathcal D}\right\rangle= \frac{1}{4}\sigma_1,
\end{equation}
which, by redefining $\varepsilon_1=-\sigma_0$, $\varepsilon_2=-\sigma_0+\sigma_1$, can be rewritten in terms of the generators $\boldsymbol{X}$ and $\boldsymbol{X}_{ab}$ of \eqref{sl2xu1v2} as
\begin{equation}\label{invtsl2u1}
 \left\langle \boldsymbol{X},  \boldsymbol{X} \right\rangle= \varepsilon_1
,\quad
\left\langle \boldsymbol{X}_{ab} ,  \boldsymbol{X}_{cd} \right\rangle= -\varepsilon_1  \delta _{a(c}\delta _{d)b}+\varepsilon_ 2 \delta _{ab}\delta _{cd}.
\end{equation}
Thus, we will consider the invariant tensor on $\mathfrak{sl}(3,\mathbb R)\oplus \mathfrak{sl}(2,\mathbb R)\oplus \mathfrak{so}(2)$ defined by \eqref{invtsl3} and \eqref{invtsl2u1}. Using the redefinitions \eqref{gencont2}, \eqref{redefitsltwu1} and $\gamma_2=\varepsilon_2-\frac23 \varepsilon_0$, the limit $c\rightarrow\infty$ yields an invariant bilinear form on the $\mathfrak{hs_3 nw2}$ algebra \eqref{HSNWalgebra2}, given by \eqref{itnw} plus the relations
\begin{equation}\label{invtcont2}
\begin{aligned}
&\left\langle \tilde{\boldsymbol{\mathcal K}}_{ab} , \tilde{\boldsymbol{\Sigma}}_{cd}\right\rangle= \gamma_0 \left( \delta _{a(c}\delta _{d)b}-\frac{2}{3}\delta _{ab}\delta _{cd}\right)
,\\
&\left\langle \tilde{\boldsymbol{\mathcal K}}_{ab} ,  \tilde{\boldsymbol{\mathcal K}}_{cd} \right\rangle= -\gamma_1 \delta _{a(c}\delta _{d)b}+\gamma_2 \delta _{ab}\delta _{cd}  ,\quad
\left\langle \tilde{\boldsymbol{\mathcal K}}_a , \tilde{\boldsymbol{\mathcal K}}_b \right\rangle=- \gamma_0  \delta_{ab}.
\end{aligned}
\end{equation}
Again, this invariant tensor has non-vanishing determinant as long as $\gamma_0\neq0$. If, instead of $\tilde{\boldsymbol{\Sigma}}_{ab}$, one uses the generator $\tilde{\boldsymbol{\Sigma}}^{\prime}_{ab}$ defined in \eqref{redefSigma}, the first relation in \eqref{invtcont2} is modified as
\begin{equation}\label{invtcont2prime}
\left\langle \tilde{\boldsymbol{\mathcal K}}_{ab} , \tilde{\boldsymbol{\Sigma}}^{\prime}_{cd}\right\rangle= \gamma_0 \delta _{a(c}\delta _{d)b}.
\end{equation}

\end{itemize}
The next step is to use these results to define non-degenerate invariant bilinear forms on the expanded algebras $S\times\mathfrak{hs_3 nw1}$ and  $S\times\mathfrak{hs_3 nw2}$ given by \eqref{HSNWalgebra1exp} and \eqref{HSNWalgebra2exp}, respectively. Following \cite{Boulanger:2002bt,Izaurieta:2006zz}, the procedure is summarized as follows: given a set of Lie algebra generators $T_\alpha$ together with an invertible invariant bilinear form $\langle T_\alpha T_\beta \rangle = g_{\alpha\beta}$, the expanded Lie algebra spanned by the generators $T^{(i)}_\alpha= \lambda_i \times T_\alpha$ admits the non-degenerate invariant tensor  $\langle T^{(i)}_\alpha T^{(j)}_\beta \rangle = h_{ij} g_{\alpha\beta}$, where the components of $h_{ij}=\langle \lambda_i \lambda_j \rangle$ include a set of arbitrary constants. When considering the $S\times \mathfrak{nw}$ algebra. \eqref{expNW}, we find \cite{Penafiel:2019czp}
\begin{equation}\label{itnwexp}
\left \langle \boldsymbol{k}^{(i)},\boldsymbol{\sigma}^{(j)} \right \rangle = - \mu_{i\diamond j} ,
\quad \left \langle \boldsymbol{k}_a^{(i)},
\boldsymbol{k}_b^{(j)} \right \rangle = \mu_{i\diamond j}\delta_{ab} ,
\quad \left \langle \boldsymbol{k}^{(i)},\boldsymbol{k}^{(j)} \right
\rangle = \nu_{i\diamond j},
\end{equation}
where $\mu_{i}$ and $\nu_{i}$ stand for arbitrary constants. The procedure can be generalized straightforwardly to the case of the $S\times \mathfrak{hs_3 nw1}$ and $S\times \mathfrak{hs_3 nw2}$ algebras:
\begin{itemize}

\item The generators of $S\times \mathfrak{hs_3 nw1}$ are given by \eqref{expandedgen} and \eqref{expandedgenhs1}. Thus, we can use the results \eqref{itnw} and \eqref{invtcont1} to obtain the invariant tensor \eqref{itnwexp} together with the extension
\begin{equation}\label{itexp1}
\begin{aligned}
&\left\langle \boldsymbol{\mathcal K}_a^{(i)}  , \boldsymbol{\Sigma}_b^{(j)} \right\rangle= -\mu_{i\diamond j} \delta_{ab} ,
\\
&\left\langle \boldsymbol{\mathcal K}_{ab}^{(i)} ,  \boldsymbol{\mathcal K}_{cd}^{(j)} \right\rangle= \mu_{i\diamond j} \left( \delta _{a(c}\delta _{d)b}-\frac{2}{3}\delta _{ab}\delta _{cd} \right)  ,
\\
&\left\langle \boldsymbol{\mathcal K}_a^{(i)}  ,\boldsymbol{\mathcal K}_b^{(j)} \right\rangle= \nu_{i\diamond j}  \delta_{ab}.
\end{aligned}
\end{equation}

\item For $S\times \mathfrak{hs_3 nw2}$ we use the invariant form defined by \eqref{itnw} and \eqref{invtcont2} to compute an invariant bilinear form for the expanded generators \eqref{expandedgen} and \eqref{expandedgenhs2}. After the expansion, the invariant bilinear form \eqref{itnwexp} is extended to include the relations
\begin{equation}\label{itexp2}
\begin{aligned}
&\left\langle \tilde{\boldsymbol{\mathcal K}}_{ab}^{(i)} , \tilde{\boldsymbol{\Sigma}}_{cd}^{(j)}\right\rangle= \mu_{i\diamond j} \left( \delta _{a(c}\delta _{d)b}-\frac{2}{3}\delta _{ab}\delta _{cd}\right)
,\\
&\left\langle \tilde{\boldsymbol{\mathcal K}}_{ab}^{(i)} ,  \tilde{\boldsymbol{\mathcal K}}_{cd}^{(j)} \right\rangle= -\nu_{i\diamond j}  \delta _{a(c}\delta _{d)b}+\rho_{i\diamond j}  \delta _{ab}\delta _{cd} ,
\\
&\left\langle \tilde{\boldsymbol{\mathcal K}}_a^{(i)} , \tilde{\boldsymbol{\mathcal K}}_b^{(j)} \right\rangle=- \mu_{i\diamond j}  \delta_{ab}.
\end{aligned}
\end{equation}
As shown for \eqref{invtcont2prime}, one can simplify the first relation in \eqref{itexp2} to
\begin{equation}\label{invtcont2primeexp}
\left\langle \tilde{\boldsymbol{\mathcal K}}_{ab}^{(i)} , \tilde{\boldsymbol{\Sigma}}^{\prime(j)}_{cd}\right\rangle= \mu_{i\diamond j} \delta _{a(c}\delta _{d)b}.
\end{equation}
by redefining the expanded generator $\tilde{\boldsymbol{\Sigma}}_{ab}^{(i)}$ as in \eqref{redefSigmaexp}.

\end{itemize}

\subsection{Chern-Simons actions}
CS forms provide actions for topological field theories in $2+1$ dimensions. In the case of NR symmetries, CS theories have been used to define three-dimensional versions of Newton-Cartan gravity and extensions thereof \cite{Papageorgiou:2009zc,Bergshoeff:2016lwr,Hartong:2016yrf,Matulich:2019cdo}. In our case, after having defined an non-degenerate invariant bilinear form on the expanded spin-$3$ algebras $S\times \mathfrak{hs_3 nw1}$ and $S\times \mathfrak{hs_3 nw2}$, it is natural to construct topological higher-spin gravity actions by considering the integral of a CS three-form over a three-dimensional manifold $M_3$
\begin{equation}\label{CSform}
I=\int_{M_3} \mathcal L,\qquad \mathcal L= \left\langle \boldsymbol{A}\,\left( \boldsymbol{F} -\frac{1}{6} \left[\boldsymbol{A}, \boldsymbol{A}\right] \right)\right\rangle.
\end{equation}
where $\boldsymbol{A}$ is a connection one form taking values on the Lie algebra of interest and $\boldsymbol{F}=\mathrm{d} \boldsymbol{A} + \boldsymbol{A}^2$ is the corresponding curvature two-form\footnote{Throughout the analysis, wedge product between forms is understood.}. As done in previous sections, we start by briefly reviewing the pure Nappi-Witten case, without higher-spin fields. The gauge connection one-form on the $\mathfrak{nw}$ algebra \eqref{nwalg} read
\begin{equation}\label{NWAF}
\boldsymbol{A}^{\mathfrak{nw}}= \theta^a \boldsymbol{k}_a + \alpha \boldsymbol{k} +\beta \boldsymbol{\sigma},\quad \boldsymbol{F}^{\mathfrak{nw}}= \bar R^a \boldsymbol{k}_a + \bar R \boldsymbol{k} + \bar F \boldsymbol{\sigma},
\end{equation}
where the curvature components are given by
\begin{equation}\label{NWcurv}
\bar R^a=\mathrm{d}\theta^a+\epsilon^{ab}\,\theta^b\,\alpha,\quad
\bar R= \mathrm{d} \alpha,\quad
\bar F= \mathrm{d} \beta -\frac12\epsilon_{ab}\,\theta^a\,\theta^b.
\end{equation}
Using the invariant form \eqref{itnw}, the CS action invariant under the $\mathfrak{nw}$ algebra takes the form
\begin{equation}\label{CSNW}
\mathcal{L}_{\mathfrak{nw}}=\gamma_0\left(  \theta^a \, \bar R^a-\alpha\, \mathrm{d}\beta-\beta\, \mathrm{d}\alpha\right) +\gamma_1\,\alpha \, \mathrm{d}\alpha .
\end{equation}
The next step is to generalize this action to the case of the expanded Nappi-Wiiten algebra $S\times\mathfrak{nw}$ given in Eq. \eqref{expNW} is to introduce the semigroup elements \eqref{slaw} through the expanded generators \eqref{expandedgen}. The connection and the curvatures for $S\times\mathfrak{nw}$ are
\begin{equation}\label{expNWAF}
\boldsymbol{A}^{S\times\mathfrak{nw}}= \theta^a_{(i)} \boldsymbol{k}_a^{(i)} + \alpha_{(i)} \boldsymbol{k}^{(i)} +\beta_{(i)} \boldsymbol{\sigma}^{(i)},\quad \boldsymbol{F}^{S\times\mathfrak{nw}}= \bar R^a_{(i)} \boldsymbol{k}_a^{(i)} + \bar R_{(i)} \boldsymbol{k}^{(i)} + \bar F_{(i)} \boldsymbol{\sigma}^{(i)}\\
\end{equation}
where we have defined the expanded curvature components
\begin{equation}\label{expNWFcomp}
\bar R^a_{(i)}=\mathrm{d}\theta^a_{(i)}+\epsilon^{ab}\,\theta^b_{(j)}\,\alpha_{(k)}\,\delta^{j\diamond k}_{i},\quad
\bar R_{(i)}= \mathrm{d} \alpha_{(i)},\quad
\bar F_{(i)}= \mathrm{d} \beta_{(i)} -\frac12\epsilon_{ab}\,\theta^a_{(j)}\,\theta^b_{(k)}\,\delta^{j\diamond k}_{i},
\end{equation}
and now Einstein summation convention also includes the semigroup indices. After implementing the expansion mechanism, the CS action \eqref{CSNW} is extended to \cite{Penafiel:2019czp}
\begin{equation}\label{CSNWexp}
\mathcal{L}_{S\times\mathfrak{nw}}=\mu_{i\diamond j}\left(  \theta^a_{(i)} \, \bar R^a_{(j)}-\alpha_{(i)}\, \mathrm{d}\beta_{(j)}-\beta_{(i)}\, \mathrm{d}\alpha_{(j)}\right) +\nu_{i\diamond j}\,\alpha_{(i)} \, \mathrm{d}\alpha_{(j)} \,,
\end{equation}
where the constants $\gamma_0$ and $\gamma_1$ are replaced by the corresponding ones entering in the expanded invariant tensor \eqref{itnwexp}, $\mu_{i}$ and $\nu_{i}$.
\\

\noindent\textbf{Spin-$3$ generalization}: This procedure previously presented can be further generalized to define CS actions invariant under the spin-$3$ extension of the Nappi-Witten algebras presented in Section \ref{HSNW}. In order to simplify the discussion, we make a parallel construction for both symmetries $\mathfrak{hs_3 nw1}$ and $\mathfrak{hs_3 nw2}$. We thus consider two connection one-forms, each one taking values on one of the spin-$3$ extensions of the Nappi-Witten algebra
\begin{equation}\label{Aexpnw}
\begin{aligned}
\boldsymbol{A}^{\mathfrak{hs_3 nw1}}&= \boldsymbol{A}^{\mathfrak{nw}}+ \Theta^{ab}\boldsymbol{\mathcal K}_{ab}+A^a \boldsymbol{\mathcal K}_a +B^a \boldsymbol{\Sigma}_a,
\\[6pt]
\boldsymbol{A}^{\mathfrak{hs_3 nw2}}&=\boldsymbol{A}^{\mathfrak{nw}} + \Theta^a\tilde{\boldsymbol{\mathcal K}}_a+A^{ab} \tilde{\boldsymbol{\mathcal K}}_{ab} +B ^{ab} \tilde{\boldsymbol{\Sigma}}_{ab},
\end{aligned}
\end{equation}
with associated curvatures
\begin{equation}
\begin{aligned}
\boldsymbol{F}^{\mathfrak{hs_3 nw1}}&= R^a \boldsymbol{k}_a+R \boldsymbol{k}+ F \boldsymbol{\sigma}
+ \mathcal R^{ab} \boldsymbol{\mathcal K}_{ab}
+\mathcal R^a \boldsymbol{\mathcal K}_a +\mathcal F^a \boldsymbol{\Sigma}_a,
\\[6pt]
\boldsymbol{F}^{\mathfrak{hs_3 nw2}}&= \tilde{R}^a \boldsymbol{k}_a+\tilde{R} \boldsymbol{k}+ \tilde{F} \boldsymbol{\sigma}
+\tilde{\mathcal R}^a \tilde{\boldsymbol{\mathcal K}}_a  + \tilde{\mathcal R}^{ab} \tilde{\boldsymbol{\mathcal K}}_{ab}+\tilde{\mathcal F}^{ab} \tilde{\boldsymbol{\Sigma}}_{ab}.
\end{aligned}
\end{equation}
Notice that, since the higher-spin generators in each expanded algebra have different index structure, the corresponding higher-spin gauge field content is different in each case as well. This manifests in the form of the components of the curvatures. Indeed, for $\boldsymbol{F}^{\mathfrak{hs_3 nw1}}$ we find
\begin{equation}
\begin{aligned}
&R^a= \bar R^a
+ 2\epsilon^{ab} \Theta^{bc}\, A^c,
\\
&R=\bar R +\frac 12 \epsilon_{ab} A^a\, A^b,
\\
&F=\bar F +\epsilon_{ab}\left(A^a B^b-2\Theta^{ac}\,\Theta^{cb}\right),
\\
&\mathcal R^{ab} = \mathrm{d} \Theta^{ab}
+
\epsilon^{c(a} \left(\frac12
 A^{b)}\, \theta^c
-\Theta^{b)c}\,\alpha
\right)- \delta^{ab}\epsilon_{cd}\theta^c \, A^d,
\\
&\mathcal R^a = \mathrm{d} A^a + \epsilon^{ab} A^b\, \alpha,
\\
&\mathcal F^a = \mathrm{d} B^a +  \epsilon^{ab} \left(B^b\,
\alpha+ A^b\, \beta\right)-2\epsilon_{bc}\Theta^{ab}\,\theta^c,
\end{aligned}
\end{equation}
whereas the components of the curvature $\boldsymbol{F}^{\mathfrak{hs_3 nw2}}$ are
\begin{equation}
\begin{aligned}
&\tilde{R}^a= \bar R^a
+2\epsilon^{ab} A^{bc}\,\Theta^c,
\\
&\tilde{R}=\bar R -2\epsilon_{ab} A^{ac}\, A^{cb},
\\
&\tilde{F}=\bar F +\epsilon_{ab}\left(\frac12\Theta^a\,\Theta^b-4A^{ac} B^{cb}\right),
\\
&\tilde{\mathcal R}^a = \mathrm{d} \Theta^a
+\epsilon^{ab} \Theta^b\, \alpha -2\epsilon_{bc} A^{ab} \, \theta^c,
\\
&\tilde{\mathcal R}^{ab} = \mathrm{d} A^{ab}
-
\epsilon^{c(a}  A^{b)c}\,\alpha,
\\
&\tilde{\mathcal F}^{ab} = \mathrm{d} B^{ab}  +\epsilon^{c(a}\left( \frac12
 \Theta^{b)}\, \theta^c -A^{b)c}\,\beta -B^{b)c}\,\alpha\right)
- \delta^{ab}\epsilon_{cd}\theta^c \, \Theta^d.
\end{aligned}
\end{equation}
Naturally, both curvatures possess the same terms $\bar R^a$, $\bar R$ and $\bar F$, defined in Eq. \eqref{NWcurv}, associated to the Nappi-Witten subalgebra that both spin-$3$ symmetries share. The CS forms invariant under $\mathfrak{hs_3 nw1}$ and $\mathfrak{hs_3 nw2}$ can be computed by evaluating \eqref{CSform} inr the corresponding conections and curvatures and later using the invariant tensors given in \eqref{invtcont1} and \eqref{invtcont2}. This yields two inequivalent spin-$3$ extensions of the Nappi-Witten CS action \eqref{CSNW},
\begin{equation}\label{CShs}
\begin{aligned}
\mathcal L_{\mathfrak{hs_3 nw1}}&= \mathcal L_{\mathfrak{nw}}
+ \gamma_0\left(2\Theta^{ab} \, \mathcal R^{ab}-\frac23 \Theta^{aa} \,  \mathcal R^{bb} -A^a \, \mathcal F^a-B^a \,  \mathcal R^a \right)+\gamma_1 A^a \, \mathcal R^a,
\\[6pt]
\mathcal L_{\mathfrak{hs_3 nw2}}&= \mathcal L_{\mathfrak{nw}}
+ \gamma_0\left(2A^{ab} \,  \tilde{\mathcal F}^{ab}+2B^{ab}  \, \tilde{\mathcal R}^{ab}-
\frac23 A^{aa}  \, \tilde{\mathcal F}^{bb}-\frac23 B^{aa} \,  \tilde{\mathcal R}^{bb}
-\Theta^a \,  \tilde{\mathcal R}^a\right)
\\&- 2\gamma_1 A^{ab} \, \tilde{\mathcal R}^{ab}
+\gamma_2 A^{aa} \, \tilde{\mathcal R}^{bb}.
\end{aligned}
\end{equation}

\noindent\textbf{Expansion}: Now we turn our attention to the expanded algebras $S\times\mathfrak{hs_3 nw1}$ and $S\times\mathfrak{hs_3 nw2}$. In order to obtain the corresponding expanded CS actions, we follow the same procedure as in the $\mathfrak{nw}$ case. Therefore, we define two expanded connections
\begin{equation}\label{expAs}
\begin{aligned}
\boldsymbol{A}^{S\times\mathfrak{hs_3 nw1}}&=\boldsymbol{A}^{S\times\mathfrak{nw}}+ \Theta_{(i)}^{ab}\boldsymbol{\mathcal K}_{ab}^{(i)}+A^a_{(i)}\boldsymbol{\mathcal K}_a^{(i)}+B_{(i)}^a \boldsymbol{\Sigma}_a^{(i)},
\\[6pt]
\boldsymbol{A}^{S\times\mathfrak{hs_3 nw2}}&=\boldsymbol{A}^{S\times\mathfrak{nw}}+ \Theta_{(i)}^a\tilde{\boldsymbol{\mathcal K}}_a^{(i)}+A^{ab}_{(i)}\tilde{\boldsymbol{\mathcal K}}_{ab}^{(i)}+B_{(i)}^{ab} \tilde{\boldsymbol{\Sigma}}_{ab}^{(i)},
\end{aligned}
\end{equation}
where $\boldsymbol{A}^{S\times\mathfrak{nw}}$ is the connection that takes values on the Expanded Nappi-Witten algebra \eqref{expNWAF}. The corresponding curvatures have the form
\begin{equation}\label{expFs}
\begin{aligned}
\boldsymbol{F}^{S\times\mathfrak{hs_3 nw1}}&= R^a_{(i)} \boldsymbol{k}_a^{(i)}+R_{(i)} \boldsymbol{k}^{(i)}+ F_{(i)} \boldsymbol{\sigma}^{(i)}
+ \mathcal R^{ab}_{(i)} \boldsymbol{\mathcal K}_{ab}^{(i)}
+\mathcal R^a_{(i)} \boldsymbol{\mathcal K}_a^{(i)} +\mathcal F^a_{(i)} \boldsymbol{\Sigma}_a^{(i)},
\\[6pt]
\boldsymbol{F}^{S\times\mathfrak{hs_3 nw2}}&= \tilde{R}^a_{(i)} \boldsymbol{k}_a^{(i)}+\tilde{R}_{(i)} \boldsymbol{k}^{(i)}+ \tilde{F}_{(i)} \boldsymbol{\sigma}^{(i)}
+\tilde{\mathcal R}^a_{(i)} \tilde{\boldsymbol{\mathcal K}}_a^{(i)}  + \tilde{\mathcal R}^{ab}_{(i)} \tilde{\boldsymbol{\mathcal K}}_{ab}^{(i)}+\tilde{\mathcal F}^{ab}_{(i)} \tilde{\boldsymbol{\Sigma}}_{ab}^{(i)},
\end{aligned}
\end{equation}
where the components of the curvature two-form on $S\times\mathfrak{hs_3 nw1}$ are given by
\begin{equation}\label{expNWFcomp1}
\begin{aligned}
&R^a_{(i)}= \bar R^a_{(i)}
+ 2\epsilon^{ab} \Theta^{bc}_{(j)} \, A^c_{(k)}
\,\delta^{j\diamond k}_{i},
\\
&R_{(i)}=\bar R_{(i)} +\frac 12 \epsilon_{ab} A^a_{(j)} \, A^b_{(k)}
\,\delta^{j\diamond k}_{i},
\\
&F_{(i)}=\bar F_{(i)} +\epsilon_{ab}\left(A^a_{(j)}  \, B^b_{(k)} -2\Theta^{ac}_{(j)} \,\Theta^{cb}_{(k)} \right)
\delta^{j\diamond k}_{i},
\\
&\mathcal R^{ab}_{(i)} = \mathrm{d} \Theta^{ab}_{(i)}
+
\epsilon^{c(a} \left(\frac12
 A^{b)}_{(j)} \, \theta^c_{(k)}
-\Theta^{b)c}_{(j)} \,\alpha_{(k)}
\right)
\delta^{j\diamond k}_{i}
- \delta^{ab}\epsilon_{cd}\theta^c_{(j)} \, A^d_{(k)}
\,\delta^{j\diamond k}_{i},
\\
&\mathcal R^a_{(i)} = \mathrm{d} A^a_{(i)} + \epsilon^{ab} A^b_{(j)} \, \alpha_{(k)}
\,\delta^{j\diamond k}_{i},
\\
&\mathcal F^a_{(i)} = \mathrm{d} B^a_{(i)} +  \epsilon^{ab} \left(B^b_{(j)} \,
\alpha_{(k)} + A^b_{(j)} \, \beta_{(k)} \right)\delta^{j\diamond k}_{i}
-2\epsilon_{bc}
\Theta^{ab}_{(j)} \,\theta^c_{(k)}
\,\delta^{j\diamond k}_{i},
\end{aligned}
\end{equation}
while for the components of the curvature on the second expanded algebra $\boldsymbol{F}^{S\times\mathfrak{hs_3 nw2}}$ we get
\begin{equation}\label{expNWFcomp2}
\begin{aligned}
&\tilde{R}^a_{(i)}= \bar R^a_{(i)}
+2\epsilon^{ab} A^{bc}_{(j)} \,\Theta^c_{(k)}
\,\delta^{j\diamond k}_{i},
\\
&\tilde{R}_{(i)}=\bar R_{(i)} -2\epsilon_{ab} A^{ac}_{(j)} \, A^{cb}_{(k)}
\,\delta^{j\diamond k}_{i},
\\
&\tilde{F}_{(i)}=\bar F_{(i)} +\epsilon_{ab}\left(\frac12\Theta^a_{(j)} \,\Theta^b_{(k)} -4A^{ac}_{(j)}  B^{cb}_{(k)} \right)
\delta^{j\diamond k}_{i},
\\
&\tilde{\mathcal R}^a_{(i)} = \mathrm{d} \Theta^a_{(i)}
+\epsilon^{ab} \Theta^b_{(j)} \, \alpha_{(k)}  \,\delta^{j\diamond k}_{i}
-2\epsilon_{bc} A^{ab}_{(j)}  \, \theta^c_{(k)}
\,\delta^{j\diamond k}_{i},
\\
&\tilde{\mathcal R}^{ab}_{(i)} = \mathrm{d} A^{ab}_{(i)}
-
\epsilon^{c(a}  A^{b)c}_{(j)} \,\alpha_{(k)}
\,\delta^{j\diamond k}_{i},
\\
&\tilde{\mathcal F}^{ab}_{(i)} = \mathrm{d} B^{ab}_{(i)}  +\epsilon^{c(a}\left( \frac12
 \Theta^{b)}_{(j)} \, \theta^c_{(k)}  -A^{b)c}_{(j)} \,\beta_{(k)}  -B^{b)c}_{(j)} \,\alpha_{(k)} \right)\delta^{j\diamond k}_{i}
- \delta^{ab}\epsilon_{cd}\theta^c_{(j)}  \,
\Theta^d_{(k)}
\,\delta^{j\diamond k}_{i}.
\end{aligned}
\end{equation}
Note that, as before, the components of the curvature two-forms along the expanded generators $\boldsymbol{k}_a^{(i)}$, $\boldsymbol{k}^{(i)}$ and $\boldsymbol{\sigma}^{(i)}$ contain the terms given in \eqref{expNWFcomp} plus contributions of the higher-spin fields that are different in each case due to the structure of the commutation relations involving the higher-spin generators. The expanded CS actions can be obtained in a straightforward way by applying the same procedure used for the action \eqref{CSNW} to obtain \eqref{CSNWexp}: i) take the actions given in \eqref{CShs} and replace $\gamma_0\rightarrow\mu_{i\diamond j}$, $\gamma_1\rightarrow\nu_{i\diamond j}$, $\gamma_2\rightarrow\rho_{i\diamond j}$, ii) insert the semigroup indices $i$ and $j$ in the gauge fields and curvatures, contracted with the invariant bilinear form constants in the proper way, iii) take care of using Kronecker deltas of the form $\delta^{j\diamond k}_{i}$ in the terms that come from the commutator $[\boldsymbol{A},\boldsymbol{A}]$. We find the resulting actions
\begin{equation}\label{hsCS1}
\begin{aligned}
\mathcal L_{S\times\mathfrak{hs_3 nw1}}=\; \mathcal L_{S\times\mathfrak{nw}}
+ \mu_{i\diamond j}\left(2\Theta^{ab}_{(i)} \, \mathcal R^{ab}_{(j)}-\dfrac23 \Theta^{aa}_{(i)} \, \mathcal R^{bb}_{(j)} -A^a_{(i)} \, \mathcal F^a_{(j)}-B^a_{(i)} \, \mathcal R^a_{(j)} \right)
+ \nu_{i\diamond j} \;A^a_{(i)} \,  \mathcal R^a_{(j)},
\end{aligned}
\end{equation}
\begin{equation}\label{hsCS2}
\begin{aligned}
\mathcal L_{S\times\mathfrak{hs_3 nw2}}&=\; \mathcal L_{S\times\mathfrak{nw}}
+  \mu_{i\diamond j} \left(2A^{ab}_{(i)}  \, \tilde{\mathcal F}^{ab}_{(j)}+2B^{ab}_{(i)}  \, \tilde{\mathcal R}^{ab}_{(j)}-
\dfrac23 A^{aa}_{(i)} \,  \tilde{\mathcal F}^{bb}_{(j)}-\dfrac23 B^{aa}_{(i)} \,  \tilde{\mathcal R}^{bb}_{(j)}
-\Theta^a_{(i)}  \, \tilde{\mathcal R}^a_{(j)}\right)
\\[10pt]
&\hskip.5truecm- 2 \nu_{i\diamond j} \;A^{ab}_{(i)} \, \tilde{\mathcal R}^{ab}_{(j)}
+ \rho_{i\diamond j} \;A^{aa}_{(i)} \, \tilde{\mathcal R}^{bb}_{(j)},
\end{aligned}
\end{equation}
where one should remark that, in the same way as for the spatial indices, there is an implicit sum over repeated semigroup indices $i$ and $j$. Naturally, these actions can alternatively be obtained by evaluating \eqref{CSform} using the connections and curvatures given in \eqref{expAs} and \eqref{expFs} together with the invariant bilinear forms given in Eqs. \eqref{itnwexp}-\eqref{itexp2}. As we have seen, however, once the CS action invariant under some to-be-expanded algebra is known, the expansion method provides a quite efficient and direct way to obtain the CS actions invariant under the corresponding expanded algebra.

\subsection{Non-relativistic limit of expanded $\mathfrak{sl}(3,\mathbb R)$ algebras}
\label{NRlim}
It is natural to expect that the NR spin-$3$ symmetries $S\times \mathfrak{hs_3 nw1}$ and $S\times \mathfrak{hs_3 nw2}$, respectively defined by \eqref{HSNWalgebra1exp} and \eqref{HSNWalgebra2exp}, are not only expansions of NR algebra, but also NR contractions of some relativistic symmetry. In \cite{Caroca:2017izc}, spin-$3$ extensions of relativistic algebras in $2+1$ dimensions were obtained as expansions of $\mathfrak{sl}(3,\mathbb R)$. Combining the notation introduced in \cite{Caroca:2017izc} and the definition \eqref{slaw}, an expanded algebra of the form $S\times \mathfrak{sl}(3,\mathbb R)$ has the commutation relations
\begin{equation}\label{sl3exp}
\left[\boldsymbol{L}_A^{(i)},\boldsymbol{L}_B^{(j)}\right] =\epsilon_{AB}^{\quad C}\boldsymbol{L}_{C}^{(i\diamond j)}\,\quad
\left[\boldsymbol{L}_A^{(i)},\boldsymbol{\mathcal L}_{BC}^{(j)}\right] =\epsilon_{A(B}^{\quad \;D}\boldsymbol{\mathcal L}_{C)D}^{(i\diamond j)}\,\quad
\left[\boldsymbol{\mathcal L}_{AB}^{(i)},\boldsymbol{\mathcal L}_{CD}^{(j)}\right]  =- \eta_{(A(C}\epsilon_{D)B)}^{\quad \;E} \boldsymbol{L}_E^{(i\diamond j)}.
\end{equation}
Depending on the chosen semigroup, this structure can reproduce spin-$3$ extensions of the Ponicar\'e, AdS, Maxwell and AdS-Lorentz algebras (see \eqref{SMSEhsrel}). The method was also extended to define spin-$3$ extensions of the Generalized Poincar\'e  (also known as $\mathfrak B_m$) \cite{Izaurieta:2009hz} and Generalized AdS (also known as $\mathfrak C_m$) \cite{Concha:2016kdz} algebras in three space-time dimensions.

It is interesting to point out that the contractions introduced in Section \ref{NRcontractions} reveal the way to extend \eqref{sl3exp} in order to define expanded relativistic symmetries whose NR limits are precisely the expanded algebras $S\times \mathfrak{hs_3 nw1}$ and $S\times \mathfrak{hs_3 nw2}$.

\begin{itemize}
\item By expanding the $\mathfrak{so}(3)$ algebra \eqref{so3alg} we find
\begin{equation}\label{so3exp}
\begin{aligned}
\left[\boldsymbol{X}^{(i)},\boldsymbol{X}_a^{(j)}\right]=&\epsilon_{ab}\boldsymbol{X}_b^{(i\diamond j)}
,\qquad
\left[\boldsymbol{X}_a^{(i)}, \boldsymbol{X}_b^{(j)}\right]=&\epsilon_{ab} \boldsymbol{X}^{(i\diamond j)},
\end{aligned}
\end{equation}
which can be put together with \eqref{sl3exp} to define the contraction
\begin{equation}\label{expreli}
\left[S\times \mathfrak{sl}(3,\mathbb R)\right] \oplus \left[S\times \mathfrak{so}(3)\right]
\quad
\underset{c\rightarrow\infty}{\longrightarrow} \qquad S\times\mathfrak{hs_3 nw1}
\end{equation}
by means of the generator redefinition
\begin{equation}\label{redefsl3iexp}
\begin{aligned}
&\boldsymbol{L}_{0}^{(i)}= \dfrac{1}{2}\boldsymbol{k}^{(i)}+c^{2}\boldsymbol{\sigma}^{(i)},
&\quad&
\boldsymbol{L}_a^{(i)}=c\, \boldsymbol{k}_a^{(i)},
&\quad&
\boldsymbol{X}^{(i)} =\dfrac{1}{2}\boldsymbol{k}-c^{2}\boldsymbol{\sigma}^{(i)},
\\
&\boldsymbol{\mathcal L}_{a0}^{(i)} =\dfrac{1}{2}\boldsymbol{\mathcal K}_a^{(i)}+c^{2}\boldsymbol{\Sigma}_a^{(i)},
&\quad&
\boldsymbol{\mathcal L}_{ab}^{(i)} = c\,\boldsymbol{\mathcal K}_{ab}^{(i)},
&\quad&
\boldsymbol{X}_a^{(i)}=\dfrac{1}{2}\boldsymbol{\mathcal K}_a^{(i)}-c^{2}\boldsymbol{\Sigma}_a^{(i)}.
\end{aligned}
\end{equation}
In the limit $c\rightarrow \infty$ this contraction yields exactly the commutation relations \eqref{HSNWalgebra1exp} of the algebra $S\times \mathfrak{hs_3 nw1}$. What we have done here, oppositely to the \emph{first contract and later expand} procedure given in the previous sections, is to expand the relativistic relations \eqref{sl3exp} and \eqref{so3exp} before applying the contraction.

\item The expansion and the contraction also commute if we try to obtain the $S\times \mathfrak{hs_3 nw2}$ algebra. One can use \eqref{sl3exp} together with the expansion of the $\mathfrak{sl}(2,\mathbb R)\oplus \mathfrak{so}(2)$ algebra \eqref{sl2xu1v2}
\begin{equation}\label{sl2xu1exp}
\begin{aligned}
\left[\boldsymbol{X}^{(i)},\boldsymbol{X}_{ab}^{(j)}\right]=&-\epsilon_{c(a}\boldsymbol{X}_{b)c}^{(i\diamond j)}
,\qquad
\left[\boldsymbol{X}_{ab}^{(i)},\boldsymbol{X}_{cd}^{(j)}\right]=&\delta_{(a(c}\epsilon_{d)b)} \boldsymbol{X}^{(i\diamond j)}
\end{aligned}
\end{equation}
to define a NR limit
\begin{equation}\label{exprelii}
\left[S\times \mathfrak{sl}(3,\mathbb R)\right] \oplus \left[S\times \mathfrak{sl}(2,\mathbb R)\right]\oplus \left[S\times \mathfrak{so}(2) \right]
\quad
\underset{c\rightarrow\infty}{\longrightarrow} \qquad S\times\mathfrak{hs_3 nw2}.
\end{equation}
Thus, by defining relations analog to \eqref{redefrelii},
\begin{equation}\label{redefsl3iiexp}
\begin{aligned}
&\boldsymbol{L}_{0}^{(i)}= \dfrac{1}{2}\boldsymbol{k}^{(i)}+c^{2}\boldsymbol{\sigma}^{(i)},
&\quad&
\boldsymbol{L}_a^{(i)}=c\, \boldsymbol{k}_a^{(i)},
&\quad&
\boldsymbol{X}^{(i)} =\dfrac{1}{2}\boldsymbol{k}-c^{2}\boldsymbol{\sigma}^{(i)},
\\
&\boldsymbol{\mathcal L}_{ab}^{(i)} =\dfrac{1}{2}\tilde{\boldsymbol{\mathcal K}}_{ab}^{(i)}+c^{2}\tilde{\boldsymbol{\Sigma}}_{ab}^{(i)},
&\quad&
\boldsymbol{\mathcal L}_{a0}^{(i)} = c\,\tilde{\boldsymbol{\mathcal K}}_a^{(i)} ,
&\quad&
\boldsymbol{X}_{ab}^{(i)}=\dfrac{1}{2}\tilde{\boldsymbol{\mathcal K}}_{ab}^{(i)}-c^{2}\tilde{\boldsymbol{\Sigma}}_{ab}^{(i)},
\end{aligned}
\end{equation}
the limit $c\rightarrow\infty$ leads to the $S\times\mathfrak{hs_3 nw2}$ commutation relations \eqref{HSNWalgebra2exp}, es expected. This construction provides an alternative method to generate the expanded algebras that will be presented in the next section, which is based on the fact that the expansion and contraction procedures commute. The construction of the invariant bilinear forms \eqref{itnwexp}-\eqref{itexp2}, as well as the CS actions \eqref{hsCS1} and \eqref{hsCS2} that are invariant under these expanded NR symmetries can be obtained by means of this alternative method as well.

\end{itemize}

\subsection{Flat limit}
\label{flatlimit}
In the following section, we will use two particular semigroups to construct higher-spin NR symmetries, namely, the $S_E^{(n)}$ and $S_M^{(n)}$ semigroups \cite{Izaurieta:2006zz}, defined by the following multiplication rules \eqref{slaw}.
\begin{equation}\label{sen}
S_E^{(n)}=\left\{\lambda_0,\lambda_1,\dots,\lambda_{n+1} \right\},\qquad i\diamond j =\left\{
\begin{array}{lll}
i +j  \quad & \mathrm{if}\quad i+j \leq n\,,
&  \\
n+1 \quad & \mathrm{if}\quad i+j >n\,. &
\end{array}%
\right.
\end{equation}
\begin{equation}\label{smn}
S_M^{(n)}=\{\lambda_0 ,\lambda_1,\dots,\lambda_n\},\qquad
i\diamond j=\left\{
\begin{array}{lll}
i+j \,\,\,\, & \mathrm{if}\,\,\,\,\ i +j \leq n\,,
&  \\
i +j -2\left[ \frac{n+1}{2}\right]\,\,\, & \mathrm{if}%
\,\,\,\,i+j>n \,.&
\end{array}%
\right.
\end{equation}
Note that $S_E^{(n)}$ has an extra \emph{zeroth} element $\lambda_{n+1}$ ($\lambda_i\cdot\lambda_{n+1}=\lambda_{n+1}\;\forall\;\lambda_i\in S_E^{(n)}$), not present in $S_M^{(n)}$. When defining an expanded Lie algebra, the fact that the semigroup $S_E^{(n)}$ has a zero requires to impose the \emph{reduction condition} $T^{(n+1)}_\alpha\equiv0$ on the expanded generators, together with $ \mu_{n+1}\equiv0$ and  $\nu_{n+1}\equiv0$ (and additionally $\rho_{n+1}\equiv0$ for $S\times\mathfrak{hs_3 nw2}$) on the invariant tensor constants introduced in \eqref{itnwexp}-\eqref{itexp2}. This condition is not necessary when defining expanded algebras by means of the semigroup $S_M^{(n)}$.

In \cite{Penafiel:2019czp}, it was shown that the expanded Nappi-Witten algebras $S_E^{(n)}\times\mathfrak{nw}$ define \emph{Generalized Bargmann algebras}, i.e. an infinite family of extensions of the Bargmann algebra in $2+$1 dimensions. On the other hand, the expanded algebras $S_M^{(n)}\times\mathfrak{nw}$ were named  \emph{Generalized Newton-Hooke algebras} since they provide generalizations of the $S_E^{(n)}\times\mathfrak{nw}$ algebras that include a cosmological constant. By using the same semigroups in the definition of the expanded higher-spin Nappi-Witten algebra $\mathfrak{hs_3 nw1}$ and $\mathfrak{hs_3 nw2}$, we can define the following families of spin-$3$ symmetries
\begin{align}
&\textnormal{Generalized spin-}3\;\textnormal{Bargmann algebras}:&\quad&
S_E^{(n)}\times \mathfrak{hs_3 nw1},\;
S_E^{(n)}\times \mathfrak{hs_3 nw2}, \label{genbarg}\\[9pt]
&\textnormal{Generalized spin-}3\;\textnormal{Newton-Hooke algebras}:&\quad&
S_M^{(n)}\times \mathfrak{hs_3 nw1},\; \label{gennh}
S_M^{(n)}\times \mathfrak{hs_3 nw2}.
\end{align}
These symmetries are related by an In\"o\"u-Wigner contraction, which can be interpreted as a vanishing cosmological constant limit or, equivalently, as the limit when the AdS radius, say $\ell$, becomes infinite. The constant $\ell$ can be made explicit in the symmetry algebras $S^{(n)}_M\times \mathfrak{hs_3 nw1}$ and $S^{(n)}_M\times \mathfrak{hs_3 nw2}$ previously presented by rescaling the expanded generators \eqref{expandedgen}, \eqref{expandedgenhs1}, \eqref{expandedgenhs2} in the form
\begin{equation}\label{recalingsell}
\begin{aligned}
&\boldsymbol{k}_a^{(i)}\rightarrow\ell^i\, \boldsymbol{k}_a^{(i)},&\quad&
\boldsymbol{k}^{(i)}\rightarrow\ell^i \,\boldsymbol{k}^{(i)},&\quad&
\boldsymbol{\sigma}^{(i)}\rightarrow\ell^i\, \boldsymbol{\sigma}^{(i)},\\
&\boldsymbol{\mathcal K}_{ab}^{(i)}\rightarrow\ell^i \,\boldsymbol{\mathcal K}_{ab}^{(i)},&\quad&
\boldsymbol{\mathcal K}_a^{(i)}\rightarrow\ell^i \,\boldsymbol{\mathcal K}_a^{(i)},&\quad&
\boldsymbol{\Sigma}_a^{(i)}\rightarrow\ell^i \,\boldsymbol{\Sigma}_a^{(i)},\\
&\boldsymbol{\mathcal K}_a^{(i)}\rightarrow\ell^i \,\boldsymbol{\mathcal K}_a^{(i)},&\quad&
\boldsymbol{\mathcal K}_{ab}^{(i)}\rightarrow\ell^i \,\boldsymbol{\mathcal K}_{ab}^{(i)},&\quad&
\boldsymbol{\Sigma}_{ab}^{(i)}\rightarrow\ell^i\, \boldsymbol{\Sigma}_{ab}^{(i)},
\end{aligned}
\end{equation}
and similarly for the invariant tensor constants given in \eqref{itnwexp}-\eqref{itexp2}, i.e.
\begin{equation}
\mu_i\rightarrow\ell^i \mu_i,\quad \nu_i\rightarrow\ell^i \rho_i,\quad\rho_i\rightarrow\ell^i \rho_i.
\end{equation}
For a fixed value of $n$, the contraction $\ell\rightarrow\infty$ takes an expanded algebra \eqref{gennh} into the corresponding expanded algebra \eqref{genbarg}, i.e.
\begin{equation}\label{SMSEcontraction}
S_M^{(n)}\times \mathfrak{hs_3 nw1} \xrightarrow[\ell\rightarrow\infty]\, S_E^{(n)}\times \mathfrak{hs_3 nw1},\qquad
S_M^{(n)}\times \mathfrak{hs_3 nw2} \xrightarrow[\ell\rightarrow\infty]\, S_E^{(n)}\times \mathfrak{hs_3 nw2}.
\end{equation}
The rescaling \eqref{recalingsell} implies that the gauge fields in the connection one forms \eqref{Aexpnw} and \eqref{expAs} should be rescaled with the opposite powers of $\ell$,
\begin{equation}\label{rescNRfields}
\begin{aligned}
&\theta^a_{(i)}\rightarrow\dfrac{1}{\ell^i}\theta^a_{(i)},&\quad&
\alpha_{(i)}\rightarrow\dfrac{1}{\ell^i} \alpha_{(i)},&\quad&
\beta_{(i)}\rightarrow\dfrac{1}{\ell^i} \beta_{(i)},\\
&\Theta^{ab}_{(i)}\rightarrow\dfrac{1}{\ell^i}\Theta^{ab}_{(i)},&\quad&
A^a_{(i)}\rightarrow\dfrac{1}{\ell^i} A^a_{(i)},&\quad&
B^a_{(i)}\rightarrow\dfrac{1}{\ell^i} B^a_{(i)},\\
&\Theta^a_{(i)}\rightarrow\dfrac{1}{\ell^i}\Theta^a_{(i)},&\quad&
A^{ab}_{(i)}\rightarrow\dfrac{1}{\ell^i} A^{ab}_{(i)},&\quad&
B^{ab}_{(i)}\rightarrow\dfrac{1}{\ell^i} B^a_{(i)},\\
\end{aligned}
\end{equation}
and similarly for the corresponding curvatures \eqref{expFs}. In this way, one can apply the flat limit $\ell\rightarrow\infty$ to the expanded CS actions \eqref{hsCS1} and \eqref{hsCS2} and reduce higher-spin gravity actions with $S^{(n)}_M$ invariance to the corresponding $S^{(n)}_E$-invariant ones.
\\

\noindent\textbf{NR and flat limits}: As a final remark, the flat limit $\ell\rightarrow\infty$ can also be used at the level of the relativistic symmetries introduced in Section \ref{NRlim}. In particular, one can relate the NR limit given in \eqref{expreli} for $S=S^{(n)}_M$ with the corresponding ones for $S=S^{(n)}_E$.

\begin{itemize}
\item We first look at the $\mathfrak{sl}(3,\mathbb R)\oplus \mathfrak{so}(3)$ case \eqref{expreli}, where one can extend the diagram \eqref{SMSEcontraction} as
\begin{equation}\label{fulldiagram1}
\begin{aligned}
&\left[S^{(m)}_M\times \mathfrak{sl}(3,\mathbb R)\right] \oplus \left[S^{(m)}_M\times \mathfrak{so}(3)\right]
\;
&\underset{c\rightarrow\infty}{\longrightarrow}& \qquad S^{(m)}_M\times\mathfrak{hs_3 nw1}
\\
&\hskip2.3truecm {\scriptstyle \ell\rightarrow\infty}\;\Big\downarrow  & &\hskip.9truecm {\scriptstyle \ell\rightarrow\infty}\;\Big\downarrow\\
&\left[S^{(m)}_E\times \mathfrak{sl}(3,\mathbb R)\right] \oplus \left[S^{(m)}_E\times \mathfrak{so}(3)\right]
\;
&\underset{c\rightarrow\infty}{\longrightarrow}& \qquad S^{(m)}_E\times\mathfrak{hs_3 nw1}
\end{aligned}
\end{equation}
To see this, it is necessary to extend the rescaling \eqref{recalingsellRel} to the case of the relativistic algebras \eqref{expreli} and \eqref{exprelii} for $S=S^{(n)}_M$,
\begin{equation}\label{recalingsellRel}
\boldsymbol{L}_A^{(i)}\rightarrow\ell^i\, \boldsymbol{L}_A^{(i)},\quad
\boldsymbol{\mathcal L}_{AB}^{(i)}\rightarrow\ell^i \,\boldsymbol{\mathcal L}_{AB}^{(i)},
\end{equation}
and establish the following relations between relativistic spin-$3$ symmetries \cite{Caroca:2017izc}
\begin{equation}\label{SMSEhsrel}
\begin{aligned}
&S^{(1)}_M\times\mathfrak{sl}(3,\mathbb R)\simeq\mathfrak{hs_3 ads_{2+1}} &\xrightarrow[\ell\rightarrow\infty]\, &\qquad S^{(1)}_E\times\mathfrak{sl}(3,\mathbb R)\simeq\mathfrak{hs_3 poin_{2+1}}\\
&S^{(2)}_M\times\mathfrak{sl}(3,\mathbb R)\simeq\mathfrak{hs_3 adslor_{2+1}} &\xrightarrow[\ell\rightarrow\infty]\, &\qquad S^{(2)}_E\times\mathfrak{sl}(3,\mathbb R)\simeq\mathfrak{hs_3 max_{2+1}}\\
&&\vdots&\\
&S^{(n)}_M\times\mathfrak{sl}(3,\mathbb R)&\xrightarrow[\ell\rightarrow\infty]\, &\qquad S^{(n)}_E\times\mathfrak{sl}(3,\mathbb R)\\
\end{aligned}
\end{equation}
where the first line in \eqref{SMSEhsrel} relates the spin-$3$ extensions of the AdS and the Poincar\'e algebras in $2+1$ dimensions, whereas the second line does so for the spin-$3$ extensions of the Maxwell and the AdS-Lorentz algebras. However, in order to make contact with \eqref{fulldiagram1}, one has to introduce the expanded algebra \eqref{so3exp} and similarly rescale the generators of $S^{(n)}_M\times \mathfrak{so}(3)$ as
\begin{equation}\label{rescellX1}
\boldsymbol{X}^{(i)}\rightarrow\ell^i\, \boldsymbol{X}^{(i)},\quad
\boldsymbol{X}_a^{(i)}\rightarrow\ell^i\, \boldsymbol{X}_a^{(i)},\\
\end{equation}
which leads to the contractions
\begin{equation}\label{diagexpso3}
\begin{aligned}
&S^{(1)}_M\times\mathfrak{so}(3)\simeq\mathfrak{so}(4) &\xrightarrow[\ell\rightarrow\infty]\, &\qquad S^{(1)}_E\times\mathfrak{so}(3)\simeq\mathfrak{iso}(3)\\
&S^{(2)}_M\times\mathfrak{so}(3)\simeq\mathfrak{e{\textnormal -}adslor_3} &\xrightarrow[\ell\rightarrow\infty]\, &\qquad S^{(2)}_E\times\mathfrak{so}(3)\simeq\mathfrak{e{\textnormal -}max_3}\\
&&\vdots&
\\
&S^{(m)}_M\times \mathfrak{so}(3) &\xrightarrow[\ell\rightarrow\infty]\, &\qquad S^{(m)}_E\times \mathfrak{so}(3) \\
\end{aligned}
\end{equation}
where $\mathfrak{e{\textnormal -}max_3}$ and $\mathfrak{e{\textnormal -}adslor_3}$ denote the Euclidean versions of the Maxwell algebra and the AdS-Lorentz algebra in three spatial dimensions, respectively. We have related the first two examples in \eqref{diagexpso3} to known algebras since these cases will be explicitly used in the next section.

\item When considering the expansion of $\mathfrak{sl}(3,\mathbb R)\oplus \mathfrak{sl}(2,\mathbb R)\oplus \mathfrak{so}(2)$ introduced in \eqref{exprelii}, one can also combine the $c\rightarrow\infty$ and $\ell\rightarrow\infty$ limits and extend the diagram \eqref{SMSEcontraction}in the form
\begin{equation}\label{fulldiagram2}
\begin{aligned}
&\left[S^{(m)}_M\times \mathfrak{sl}(3,\mathbb R)\right] \oplus \left[S^{(m)}_M\times \mathfrak{sl}(2,\mathbb R)\right]\oplus \left[S^{(m)}_M\times \mathfrak{so}(2) \right]
\;
&\underset{c\rightarrow\infty}{\longrightarrow}& \qquad S^{(m)}_M\times\mathfrak{hs_3 nw2}
\\
&\hskip3.8truecm {\scriptstyle \ell\rightarrow\infty}\;\Big\downarrow & &\hskip.9truecm {\scriptstyle \ell\rightarrow\infty}\;\Big\downarrow
\\
&\left[S^{(m)}_E\times \mathfrak{sl}(3,\mathbb R)\right] \oplus \left[S^{(m)}_E\times \mathfrak{sl}(2,\mathbb R)\right]\oplus \left[S^{(m)}_E\times \mathfrak{so}(2) \right]
\;
&\underset{c\rightarrow\infty}{\longrightarrow}& \qquad S^{(m)}_E\times\mathfrak{hs_3 nw2}
\end{aligned}
\end{equation}
The expansion of $\mathfrak{sl}(3,\mathbb R)$ in \eqref{fulldiagram2} is again \eqref{SMSEhsrel}, which now is to be supplemented with expansions of $\mathfrak{sl}(2,\mathbb R)\oplus \mathfrak{so}(2)$. Expanding the generators of $\mathfrak{sl}(2,\mathbb R)$ introduced in \eqref{sl2xu1v1} and using the rescaling
\begin{equation}\label{rescABC}
\boldsymbol{\mathcal A}^{(i)}\rightarrow\ell^i\, \boldsymbol{\mathcal A}^{(i)},\quad
\boldsymbol{\mathcal B}^{(i)}\rightarrow\ell^i\, \boldsymbol{\mathcal B}^{(i)},\quad
\boldsymbol{\mathcal C}^{(i)}\rightarrow\ell^i\, \boldsymbol{\mathcal C}^{(i)},
\end{equation}
allows one to relate the following expanded symmetries in the flat limit $\ell\rightarrow\infty$
\begin{equation}\label{diagexpsltw}
\begin{aligned}
&S^{(1)}_M\times\mathfrak{sl}(2,\mathbb R)\simeq\mathfrak{so}(2,2) &\xrightarrow[\ell\rightarrow\infty]\, &\qquad S^{(1)}_E\times\mathfrak{sl}(2,\mathbb R)\simeq\mathfrak{isl}(2,\mathbb R)\\
&S^{(2)}_M\times\mathfrak{sl}(2,\mathbb R)\simeq\mathfrak{adslor_{2+1}} &\xrightarrow[\ell\rightarrow\infty]\,&\qquad S^{(2)}_E\times\mathfrak{sl}(2,\mathbb R)\simeq\mathfrak{max_{2+1}}\\
&&\vdots&
\\
&S^{(m)}_M\times \mathfrak{sl}(2,\mathbb R) &\xrightarrow[\ell\rightarrow\infty]\, &\qquad S^{(m)}_E\times \mathfrak{sl}(2,\mathbb R) \\
\end{aligned}
\end{equation}
Due to the isomorphism $\mathfrak{so}(2,1)\simeq \mathfrak{sl}(2,\mathbb R)$, these relations correspond to the Euclidean continuation of the ones given in \eqref{diagexpso3}. In the first case, the expanded algebras are the AdS and the Poinac\'e algebras in three space-time dimensions, whereas in the second case one finds the Maxwell and the AdS-Lorentz algebras in $2+1$ dimensions, which are known to be related by an In\"o\"u-Wigner contraction. Note that \eqref{rescABC} can be equivalently defined in the basis \eqref{sl2xu1v2} where the expanded generators \eqref{sl2xu1exp} are rescaled as
\begin{equation}\label{rescellX2}
\boldsymbol{X}^{(i)}\rightarrow\ell^i\, \boldsymbol{X}^{(i)},\quad
\boldsymbol{X}_{ab}^{(i)}\rightarrow\ell^i\, \boldsymbol{X}_{ab}^{(i)}.
\end{equation}
Finally, to complete the expansion of \eqref{sl2xu1v1}, we need to expand the $\mathfrak{so}(2)$ generator $\boldsymbol{\mathcal D}$, which simply produces $n$ copies of $\mathfrak{so}(2)$. In this case, the corresponding rescaling of $\boldsymbol{\mathcal D}^{(i)}\rightarrow\ell^i\, \boldsymbol{\mathcal D}^{(i)}$ and flat limit does notmodify the algebra, i.e.
\begin{equation}\label{expuo}
S^{(n)}_M\times\mathfrak{so}(2)\simeq S^{(n)}_E\times\mathfrak{so}(2) \simeq \mathfrak{so}(2) \oplus \underset{n\;{\rm times}}{\cdots} \oplus\mathfrak{so}(2).
\end{equation}
Putting it all together allows constructing the left-hand side if \eqref{fulldiagram2}.
\end{itemize}
The contractions here presented also relate the CS gravity theories invariant under the corresponding symmetry algebras. It is thus a direct but involved computation to obtain the CS actions invariant under the spin-$3$ NR algebras \eqref{hsCS1} and \eqref{hsCS1}, from the corresponding CS theories based on the relativistic symmetries considered in \eqref{expreli} and \eqref{expreli}, respectively, which requires to introduce rescalings in the relativistic gauge fields in complete analogy with Eq. \eqref{rescNRfields} in the NR case.
\\

\noindent\textbf{Pure Nappi-Witten case}: It is worth remarking that, since the generators $\boldsymbol{L}_A^{(i)}$ in \eqref{sl3exp} define an expansion of the $\mathfrak{sl}(2,\mathbb R)$ subalgebra of $\mathfrak{sl}(3,\mathbb R)$, a diagram of the form \eqref{diagexpsltw} is also obtained from \eqref{SMSEhsrel} when eliminating the higher-spin generators $\boldsymbol{\mathcal L}_{AB}^{(i)}$, reducing the analysis to the pure gravity case. Dropping the higher spin generators $X_a^{(i)}$ and $X_{ab}^{(i)}$ in \eqref{so3exp} and \eqref{sl2xu1exp} leaves in each case only the expanded generator $X^{(i)}$ for which the expansion can be summarized in a diagram equivalent to \eqref{expuo}. Thus, in the pure gravity case, both diagrams \eqref{fulldiagram1} and \eqref{fulldiagram2} coincide and reduce to.
\begin{equation}\label{fulldiagram2}
\begin{aligned}
&\left[S^{(m)}_M\times \mathfrak{sl}(2,\mathbb R)\right] \oplus \left[S^{(m)}_M\times \mathfrak{so}(2) \right]
\;
&\underset{c\rightarrow\infty}{\longrightarrow}& \qquad S^{(m)}_M\times\mathfrak{nw}
\\
&\hskip3.8truecm {\scriptstyle \ell\rightarrow\infty}\;\Big\downarrow & &\hskip.9truecm {\scriptstyle \ell\rightarrow\infty}\;\Big\downarrow
\\
&\left[S^{(m)}_E\times \mathfrak{sl}(2,\mathbb R)\right] \oplus \left[S^{(m)}_E\times \mathfrak{so}(2) \right]
\;
&\underset{c\rightarrow\infty}{\longrightarrow}& \qquad S^{(m)}_E\times\mathfrak{nw}
\end{aligned}
\end{equation}
This means that for a given $n$, an expanded Nappi-Witten algebra \eqref{expnw} is the NR contractions of the semidirect sum of the corresponding expanded $\mathfrak{sl}(2,\mathbb R)$ algebra and $n$ Abelian generators. As we will see in the following section, this reproduces the known contractions that lead to the Extended Bargmann algebra \cite{Bergshoeff:2016lwr} and the Extended Electric NR Maxwell algebra \cite{Aviles:2018jzw}, and extends the result to expansions with arbitrary $n$.

\section{Examples}
\label{Examples}
Let us consider explicit examples of the higher-spin symmetries presented in the previous section.
In the following, we explicitly evaluate expanded symmetries with the semigroups $S_E^{(n)}$ and $S_M^{(n)}$ for $n=1$ and $n=2$. The $n=1$ case reproduces the higher-spin extensions of the extended Bargmann and Newton-Hooke algebras in $2+1$ dimensions presented in \cite{Bergshoeff:2016soe}. The $n=2$ case yields novel higher-spin extensions of the NR Maxwell algebra and the NR AdS-Lorentz algebra in $2+1$ dimensions.
The explicit construction of the expanded algebras for $n\geq3$ can be straightforwardly done along the same lines and they will not be given here.

\subsection{Spin-$3$ Bargmann algebras}
\label{HSB}
The Bargmann algebra is the universal central extension of the Galilei algebra \cite{Bacry:1968zf} (for a recent review on non-Lorentzian symmetries, see also \cite{Bergshoeff:2022eog}). In $2+1$ dimensions, it admits a second non-trivial central extension\footnote{In fact, the Galilei algebra in three space-time dimensions admits a third central extension, which is non-physical \cite{Brihaye:1995nv} and will not be considered here.}, that allows defining a non-degenerate invariant bilinear form \cite{Papageorgiou:2009zc}. As a first example, we reconstruct the spin-$3$ extensions of the Extended Bargmann algebra in $2+1$ dimensions  \cite{Bergshoeff:2016soe}, as a particular expansion of our spin-$3$ Nappi-Witten algebras. As shown in \cite{Penafiel:2019czp}, the extended Bargmann algebra in $2+1$ dimensions
\begin{equation}\label{ExtBargAlg}
\begin{aligned}
&\left[\boldsymbol{J},\boldsymbol{G}_a\right]=\epsilon_{ab}\boldsymbol{G}_b,
&
&\left[\boldsymbol{G}_a,\boldsymbol{G}_b \right]=-\epsilon_{ab}\boldsymbol{S},\\[5pt]
&\left[\boldsymbol{J},\boldsymbol{P}_a \right]=  [\boldsymbol{H},\boldsymbol{G}_a ] =\epsilon_{ab}\boldsymbol{P}_b,
&
&\left[\boldsymbol{G}_a,\boldsymbol{P}_b \right]=-\epsilon_{ab}\boldsymbol{M},
\end{aligned}
\end{equation}
is the expanded Nappi-Witten algebra $S^{(1)}_E\times \mathfrak{nw}$, generated by
\begin{equation}\label{BargRenameGen}
\begin{aligned}
&\boldsymbol{J}=\boldsymbol{k}^{(0)},& &\boldsymbol{G}_a=\boldsymbol{k}_a^{(0)},& &\boldsymbol{S}=\boldsymbol{\sigma}^{(0)},\\
&\boldsymbol{H}=\boldsymbol{k}^{(1)},& &\boldsymbol{P}_a=\boldsymbol{k}_a^{(1)}, & &\boldsymbol{M}=\boldsymbol{\sigma}^{(1)},
\end{aligned}
\end{equation}
where the semigroup $S^{(1)}_E$ is obtained by replacing $n=1$ in \eqref{sen}. Here $\boldsymbol{J}$, $\boldsymbol{H}$, $\boldsymbol{G}_a$ and $\boldsymbol{P}_a$ stand respectively for the generators of spatial rotations, temporal translations, Galilean boosts and spatial translations, whereas $\boldsymbol{S}$ and $\boldsymbol{M}$ are central extensions associated to spin and mass\footnote{While the mass generator $\boldsymbol{M}$ is present in higher-dimensional versions of the Bargmann algebra and its physical meaning is understood (see, for example, \cite{Bacry:1968zf}), the generator $\boldsymbol{S}$ is particular to the three-dimensional case and its interpretation as a spin angular momentum generator has not yet been fully established \cite{Duval:2002cw,Hagen:2002pg}.}. In the same way, the spin-$3$ extensions of the Bargmann algebra can be defined by evaluating the expanded higher-spin symmetries \eqref{HSNWalgebra1exp} and \eqref{HSNWalgebra2exp} for $S=S_E^{(1)}$.

\begin{itemize}
\item The first spin-$3$ extension of the Extended Bargmann algebra in $2+1$ dimensions, $\mathfrak{hs_3 barg1_{2+1}}$, is given by $S_E^{(1)}\times\mathfrak{hs_3 nw1}$. It can be obtained by defining the higher-spin generators
\begin{equation}\label{hsBarg1RenameGen}
\begin{aligned}
&\boldsymbol{\mathcal J}_a=\boldsymbol{\mathcal K}_a^{(0)},&
&\boldsymbol{\mathcal G}_{ab}=\boldsymbol{\mathcal K}_{ab}^{(0)},&
&\boldsymbol{\mathcal S}_a=\boldsymbol{\Sigma}_a^{(0)}, \\
&\boldsymbol{\mathcal H}_a=\boldsymbol{\mathcal K}_a^{(1)},&
&\boldsymbol{\mathcal P}_{ab}=\boldsymbol{\mathcal K}_{ab}^{(1)},&
&\boldsymbol{\mathcal M}_a=\boldsymbol{\Sigma}_a^{(1)},
\end{aligned}
\end{equation}
and using the relations \eqref{HSNWalgebra1exp} and \eqref{sen} for $n=1$. The commutation relations of the expanded generators \eqref{BargRenameGen} and \eqref{hsBarg1RenameGen} are given by \eqref{ExtBargAlg} together with
\begin{equation}\label{BargHS1}
\begin{aligned}
&\left[\boldsymbol{J},\boldsymbol{\mathcal J}_a\right]  = \epsilon_{ab}\boldsymbol{\mathcal J}_b,
&
&\left[\boldsymbol{J},\boldsymbol{\mathcal G}_{ab}\right] =  -\epsilon_{c(a}\boldsymbol{\mathcal G}_{b)c},&
&\left[\boldsymbol{J},\boldsymbol{\mathcal S}_a\right] =\left[\boldsymbol{S},\boldsymbol{\mathcal J}_a\right]  = \epsilon_{ab}\boldsymbol{\mathcal S}_b,
\\[5pt]
&\left[\boldsymbol{G}_a,\boldsymbol{\mathcal J}_b\right] = -\epsilon_{a(c}\boldsymbol{\mathcal G}_{b)c},
&
&\left[\boldsymbol{G}_a,\boldsymbol{\mathcal G}_{bc}\right]  =  -\epsilon_{a(b}\boldsymbol{\mathcal S}_{c)}, &
&\left[\boldsymbol{\mathcal J}_a,\boldsymbol{\mathcal J}_b\right]  =  \epsilon_{ab}\boldsymbol{J},
\\[5pt]
&\left[\boldsymbol{\mathcal J}_a, \boldsymbol{\mathcal S}_b\right] = \epsilon_{ab}\boldsymbol{S},&
&\left[\boldsymbol{\mathcal J}_a,\boldsymbol{\mathcal G}_{bc}\right]  =\delta_{a(b}\epsilon_{c)d}\boldsymbol{G}_d,&
&\left[\boldsymbol{\mathcal G}_{ab},\boldsymbol{\mathcal G}_{cd}\right] = \delta_{(a(c}\epsilon_{d)b)}\boldsymbol{S},
\end{aligned}
\end{equation}
together with
\begin{equation}\label{BargHS2}
\begin{aligned}
&\left[\boldsymbol{J},\boldsymbol{\mathcal H}_a\right]=\left[\boldsymbol{H},\boldsymbol{\mathcal J}_a\right]   = \epsilon_{ab}\boldsymbol{\mathcal H}_b,&
&\left[\boldsymbol{G}_a,\boldsymbol{\mathcal H}_b\right] = \left[\boldsymbol{P}_a,\boldsymbol{\mathcal J}_b\right] = -\epsilon_{a(c}\boldsymbol{\mathcal P}_{b)c},
\\[5pt]
&\left[\boldsymbol{J},\boldsymbol{\mathcal P}_{ab}\right]=\left[\boldsymbol{H},\boldsymbol{\mathcal G}_{ab}\right]  =  -\epsilon_{c(a}\boldsymbol{\mathcal P}_{b)c},&
&\left[\boldsymbol{\mathcal J}_a,\boldsymbol{\mathcal P}_{bc}\right]  =\left[\boldsymbol{\mathcal H}_a,\boldsymbol{\mathcal G}_{bc}\right]   =\delta_{a(b}\epsilon_{c)d}\boldsymbol{P}_d,
\\[5pt]
&\left[\boldsymbol{\mathcal J}_a,\boldsymbol{\mathcal H}_b\right]  =  \epsilon_{ab}\boldsymbol{H},&
&\left[\boldsymbol{G}_a,\boldsymbol{\mathcal P}_{bc}\right]  =  \left[\boldsymbol{P}_a,\boldsymbol{\mathcal G}_{bc}\right]  =-\epsilon_{a(b}\boldsymbol{\mathcal M}_{c)},
\\[5pt]
&\left[\boldsymbol{\mathcal J}_a, \boldsymbol{\mathcal M}_b\right] =\left[\boldsymbol{\mathcal H}_a, \boldsymbol{\mathcal S}_b\right]= \epsilon_{ab}\boldsymbol{M},&
&\left[\boldsymbol{J},\boldsymbol{\mathcal M}_a\right] =\left[\boldsymbol{S},\boldsymbol{\mathcal H}_a\right]  = \left[\boldsymbol{H},\boldsymbol{\mathcal S}_a\right]=\left[\boldsymbol{M},\boldsymbol{\mathcal J}_a\right]  =\epsilon_{ab}\boldsymbol{\mathcal M}_b.
\\[5pt]
&\left[\boldsymbol{\mathcal G}_{ab},\boldsymbol{\mathcal P}_{cd}\right] = \delta_{(a(c}\epsilon_{d)b)}\boldsymbol{M},
\end{aligned}
\end{equation}
Notice that, when including the higher-spin extension, the generators $\boldsymbol{S}$ and $\boldsymbol{M}$ are not central anymore. The $\mathfrak{hs_3 barg1_{2+1}}$ algebra was obtained in \cite{Bergshoeff:2016soe} from a double contraction (large-speed-of-light and large-AdS-raduis limit) of the spin-$3$ extension of the AdS algebra in three dimensions $\mathfrak{hs_3 ads_{2+1}}$.

\item One can define a second spin-$3$ extension of the Extended Bargman algebra in $2+1$ dimensions, $\mathfrak{hs_3 barg2_{2+1}}$, as the expanded algebra $S_E^{(1)}\times \mathfrak{hs_3 nw2}$. Now, instead of \eqref{hsBarg1RenameGen}, the alternative generators
\begin{equation}\label{hsBarg2RenameGen}
\begin{aligned}
&\boldsymbol{\mathcal J}_{ab}=\tilde{\boldsymbol{\mathcal K}}_{ab}^{(0)},&
& \boldsymbol{\mathcal G}_a=\tilde{\boldsymbol{\mathcal K}}_a^{(0)},&
&\boldsymbol{\mathcal S}_{ab}=\tilde{\boldsymbol{\Sigma}}_{ab}^{(0)},& \\
&\boldsymbol{\mathcal H}_{ab}=\tilde{\boldsymbol{\mathcal K}}_{ab}^{(1)},&
&\boldsymbol{\mathcal P}_a=\tilde{\boldsymbol{\mathcal K}}_a^{(1)},&
&\boldsymbol{\mathcal M}_{ab}=\tilde{\boldsymbol{\Sigma}}_{ab}^{(1)},
\end{aligned}
\end{equation}
are introduced. Putting together the Extended Bargmann generators \eqref{BargRenameGen} with \eqref{hsBarg2RenameGen}, and using the commutation relations of the $\mathfrak{hs_3 nw2}$ algebra \eqref{HSNWalgebra2exp} for the $S_E^{(1)}$ semigroup \eqref{sen}, one finds the commutation relations \eqref{ExtBargAlg} plus
\begin{equation}\label{BargHS1T}
\begin{aligned}
&[\boldsymbol{J},\boldsymbol{\mathcal G}_a ] =\epsilon_{ab}\boldsymbol{\mathcal G}_b,&
&[\boldsymbol{J},\boldsymbol{\mathcal J}_{ab}] =-\epsilon_{c(a}\boldsymbol{\mathcal J}_{b)c},&
&[\boldsymbol{J},\boldsymbol{\mathcal S}_{ab}] =[\boldsymbol{S},\boldsymbol{\mathcal J}_{ab}] =-\epsilon_{c(a}\boldsymbol{\mathcal S}_{b)c},
\\[5pt]
&[\boldsymbol{G}_a ,\boldsymbol{\mathcal G}_b ]=-\epsilon_{a(c}\boldsymbol{\mathcal S}_{b)c},&
&[\boldsymbol{G}_a ,\boldsymbol{\mathcal J}_{bc}]=-\epsilon_{a(b}\boldsymbol{\mathcal G}_{c)},&
&[\boldsymbol{\mathcal J}_{ab},\boldsymbol{\mathcal J}_{cd}] =\delta_{(a(c}\epsilon_{d)b)} \boldsymbol{J}.
\\[5pt]
&[\boldsymbol{\mathcal G}_a ,\boldsymbol{\mathcal G}_b ] =\epsilon_{ab}\boldsymbol{S},&
&[\boldsymbol{\mathcal G}_a ,\boldsymbol{\mathcal J}_{bc}] =\delta_{a(b}\epsilon_{c)d}\boldsymbol{G}_d,&
&[\boldsymbol{\mathcal J}_{ab},\boldsymbol{\mathcal S}_{cd}] =\delta_{(a(c}\epsilon_{d)b)} \boldsymbol{S},
\end{aligned}
\end{equation}
and
\begin{equation}\label{BargHS2T}
\begin{aligned}
&[\boldsymbol{J},\boldsymbol{\mathcal P}_a ]=[\boldsymbol{H},\boldsymbol{\mathcal G}_a ] =\epsilon_{ab}\boldsymbol{\mathcal P}_b,&
&[\boldsymbol{G}_a ,\boldsymbol{\mathcal H}_{bc}]=[\boldsymbol{P}_a ,\boldsymbol{\mathcal J}_{bc}]=-\epsilon_{a(b}\boldsymbol{\mathcal P}_{c)},
\\[5pt]
&[\boldsymbol{J},\boldsymbol{\mathcal H}_{ab}] =[\boldsymbol{H},\boldsymbol{\mathcal J}_{ab}] =-\epsilon_{c(a}\boldsymbol{\mathcal H}_{b)c},&
&[\boldsymbol{\mathcal G}_a ,\boldsymbol{\mathcal H}_{bc}] =[\boldsymbol{\mathcal P}_a ,\boldsymbol{\mathcal J}_{bc}] =\delta_{a(b}\epsilon_{c)d}\boldsymbol{P}_d,
\\[5pt]
&[\boldsymbol{\mathcal J}_{ab},\boldsymbol{\mathcal H}_{cd}] =\delta_{(a(c}\epsilon_{d)b)} \boldsymbol{H},&
&[\boldsymbol{\mathcal J}_{ab},\boldsymbol{\mathcal M}_{cd}] =[\boldsymbol{\mathcal H}_{ab},\boldsymbol{\mathcal S}_{cd}] =\delta_{(a(c}\epsilon_{d)b)} \boldsymbol{M},
\\[5pt]
&[\boldsymbol{\mathcal G}_a ,\boldsymbol{\mathcal P}_b ] =\epsilon_{ab}\boldsymbol{M},&
&[\boldsymbol{J},\boldsymbol{\mathcal M}_{ab}] =[\boldsymbol{H},\boldsymbol{\mathcal S}_{ab}] =[\boldsymbol{S},\boldsymbol{\mathcal H}_{ab}]
\\[5pt]
&[\boldsymbol{G}_a ,\boldsymbol{\mathcal P}_b ]=[\boldsymbol{P}_a ,\boldsymbol{\mathcal G}_b ]=-\epsilon_{a(c}\boldsymbol{\mathcal M}_{b)c},&
&\hskip1.6truecm=[\boldsymbol{M},\boldsymbol{\mathcal J}_{ab}] =-\epsilon_{c(a}\boldsymbol{\mathcal M}_{b)c}.
\end{aligned}
\end{equation}
Even though this algebra is not explcitly equivalent to the $\mathfrak{hs_3 barg2_{2+1}}$ algebra obtained in \cite{Bergshoeff:2016soe}, by means of the Medina-Revoy theorem, one can show they are isomorphic by using the redefintion \eqref{redefSigmaexp} for the higher-spin generators $\boldsymbol{\mathcal S}_{ab}$ and $\boldsymbol{\mathcal M}_{ab}$.
\end{itemize}
Thus, we have explicitly constructed the spin-$3$ NR algebras
\begin{equation}\label{hsbiandii}
\mathfrak{hs_3 barg1_{2+1}}=S^{(1)}_E\times \mathfrak{hs_3 nw1},\qquad
\mathfrak{hs_3 barg2_{2+1}}=S^{(1)}_E\times \mathfrak{hs_3 nw2}.
\end{equation}
As argued in \cite{Bergshoeff:2016soe}, two additional higher-spin extensions of the Extended Bargmann algebra can be obtained by interchanging
\begin{equation}\label{swap1}
\begin{aligned}
&\boldsymbol{\mathcal J}_a\leftrightarrow \boldsymbol{\mathcal H}_a,&\quad&
\boldsymbol{\mathcal G}_{ab}\leftrightarrow \boldsymbol{\mathcal P}_{ab},&\quad&
\boldsymbol{\mathcal S}_a\leftrightarrow \boldsymbol{\mathcal M}_a,\\
&\boldsymbol{\mathcal J}_{ab}\leftrightarrow \boldsymbol{\mathcal H}_{ab},&\quad&
\boldsymbol{\mathcal G}_a\leftrightarrow \boldsymbol{\mathcal P}_b,&\quad&
\boldsymbol{\mathcal S}_{ab}\leftrightarrow \boldsymbol{\mathcal M}_{ab},
\end{aligned}
\end{equation}
in \eqref{BargHS1} and  \eqref{BargHS2}, and in \eqref{BargHS1T} and \eqref{BargHS2T}, respectively.
\\

\noindent\textbf{Semidirect sum structure}: The algebras $\mathfrak{hs_3 barg1_{2+1}}$ and $\mathfrak{hs_3 barg2_{2+1}}$ are semidirect sums. This is to be expected since they are higher-spin extensions of the NR limit of the Poincar\'e algebra, which is a semidirect sum itself. Indeed, the first line of \eqref{ExtBargAlg} together with Eq. \eqref{BargHS1} (Eq. \eqref{BargHS1T}), is a subalgebra isomorphic to $\mathfrak{hs_3 nw1}$ ($\mathfrak{hs_3 nw2}$). The remaining generators form an Abelian ideal that can be identified with $\mathfrak{hs_3 nw1}$ ($\mathfrak{hs_3 nw2}$) seen as an Abelian vector space, which can be denoted by $\mathfrak{hs_3 nw1}^{({\rm ab})}$ ($\mathfrak{hs_3 nw2}^{({\rm ab})}$), and on which $\mathfrak{hs_3 nw1}$ ($\mathfrak{hs_3 nw2}$) acts with the adjoint representation \cite{Barnich:2014kra}. Thus, the $\mathfrak{hs_3 barg1_{2+1}}$ ($\mathfrak{hs_3 barg2_{2+1}}$) algebra can be written in a compact form as
\begin{equation}
\left[(X,X^\ast),(Y,Y^\ast)\right]=\left([X,Y],[X,Y^\ast]-[Y,X^\ast]\right),
\end{equation}
where $X$ is an element on the $\mathfrak{hs_3 nw1}$ ($\mathfrak{hs_3 nw2}$) algebra and $X^\ast$ is an element on $\mathfrak{hs_3 nw1}^{({\rm ab})}$ ($\mathfrak{hs_3 nw2}^{({\rm ab})}$). Therefore, the spin-$3$ extensions of the Extended Bargmann algebra in $2+1$ dimensions can be defined as the semidirect sums
\begin{equation}
\begin{aligned}
\mathfrak{hs_3 barg1_{2+1}}& \simeq \mathfrak{hs_3 nw1} \oright _{\rm ad}  \mathfrak{hs_3 nw1}^{({\rm ab})},\\
\mathfrak{hs_3 barg2_{2+1}}&\simeq \mathfrak{hs_3 nw2} \oright _{\rm ad}  \mathfrak{hs_3 nw2}^{({\rm ab})}.
\end{aligned}
\end{equation}
This is the NR analog of the semidirect sum \cite{Campoleoni:2016vsh}
\begin{equation}\label{hspoin}
\mathfrak{hs_3 poin_{2+1}}\simeq\mathfrak{sl}(3,\mathbb R)\oright _{\rm ad}  \mathfrak{sl}(3,\mathbb R)^{({\rm ab})},
\end{equation}
which corresponds to the spin-$3$ extension of the Poincar\'e algebra in three space-time dimensions \cite{Afshar:2013vka,Gonzalez:2013oaa}.
\\

\noindent\textbf{NR limit}: The spin-$3$ extensions of the Bargman algebra \eqref{hsbiandii} can alternatively be obtained as In\"o\"u-Wigner contractions of relativistic symmetries. In \cite{Caroca:2017izc}, it was shown that the spin-$3$ extension Poincar\'e algebra in three space-time dimensions \eqref{hspoin} is the expansion
\begin{equation}\label{hspoin2}
\mathfrak{hs_3 poin_{2+1}}\simeq S^{(1)}_E\times \mathfrak{sl}(3,\mathbb R),
\end{equation}
with expanded generators
\begin{equation}\label{hspoingen}
\boldsymbol{J}_A^{\rm rel} = \boldsymbol{L}^{(0)}_A,\quad
\boldsymbol{P}_A^{\rm rel} = \boldsymbol{L}^{(1)}_A,\quad
\boldsymbol{\mathcal J}_{AB}^{\rm rel} = \boldsymbol{\mathcal L}^{(0)}_{AB},\quad
\boldsymbol{\mathcal P}_{AB}^{\rm rel} = \boldsymbol{\mathcal L}^{(1)}_{AB},
\end{equation}
and commutation relations given by \eqref{sl3exp} evaluated for the semigroup product law of $S^{(1)}_E$ given in \eqref{sen}. In order define take the NR limits \eqref{expreli} and \eqref{exprelii}, we also need to define the expanded algebras \eqref{so3exp} and \eqref{sl2xu1exp}. For $S=S^{(1)}_E$, these are respectively isomorphic to the Euclidean algebra in three spatial dimensions $\mathfrak{iso}(3)$ (spanned by $\{\boldsymbol{X}^{(0)},\boldsymbol{X}^{(1)},\boldsymbol{X}_a^{(0)},\boldsymbol{X}_a^{(1)}\}$ in Eq. \eqref{so3exp}) and the Poincar\'e algebra in three space-time dimensions, $\mathfrak{isl}(2,\mathbb R)$, supplemented with two Abelian generators\footnote{The isomorphism $S^{(1)}_E\times\Big( \mathfrak{sl}(2,\mathbb R)\oplus \mathfrak{so}(2)\Big) \simeq \mathfrak{isl}(2,\mathbb R)\oplus \mathfrak{so}(2)\oplus\mathfrak{so}(2)$ is more transparent at the level of the commutation relations when expanding $\mathfrak{sl}(2,\mathbb R)\oplus \mathfrak{so}(2)$ in the basis \eqref{sl2xu1v1} rather than \eqref{sl2xu1v2}.} (spanned by $\{\boldsymbol{X}^{(0)},\boldsymbol{X}^{(1)},\boldsymbol{X}_{ab}^{(0)},\boldsymbol{X}_{ab}^{(1)}\}$ in Eq. \eqref{sl2xu1exp}), i.e.
\begin{equation}\label{otherexpspoin}
\begin{aligned}
&S^{(1)}_E\times \mathfrak{so}(3) \simeq \mathfrak{iso}(3),\\
&S^{(1)}_E\times\Big( \mathfrak{sl}(2,\mathbb R)\oplus \mathfrak{so}(2)\Big) \simeq \mathfrak{isl}(2,\mathbb R)\oplus \mathfrak{so}(2)\oplus\mathfrak{so}(2).
\end{aligned}
\end{equation}
These isomorphisms also follow from the diagrams  \eqref{diagexpso3}, \eqref{diagexpsltw} and \eqref{expuo}. Inverting \eqref{redefsl3iexp}, one finds the relation between the Poincar\'e and Bargman generators
\begin{equation}\label{IWpoin1}
\begin{aligned}
\boldsymbol{J} & =  \boldsymbol{J}_{0}^{\rm rel}+\boldsymbol{X}^{(0)}, &
\boldsymbol{J}_a & =  \frac{1}{c}\boldsymbol{J}_a^{\rm rel}, &
\boldsymbol{S} & =  \frac{1}{2c^2}\left(\boldsymbol{J}_{0}^{\rm rel}-\boldsymbol{X}^{(0)}\right),\\
\boldsymbol{H} & =  \boldsymbol{P}_{0}^{\rm rel}+\boldsymbol{X}^{(1)}, &
\boldsymbol{P}_a & =  \frac{1}{c}\boldsymbol{P}_a^{\rm rel}, &
\boldsymbol{M} & =  \frac{1}{2c^2}\left(\boldsymbol{P}_{0}^{\rm rel}-\boldsymbol{X}^{(1)}\right),
\end{aligned}
\end{equation}
and the corresponding relations between NR and relativistic spin-$3$ generators
\begin{equation}\label{IWpoin2}
\begin{aligned}
\boldsymbol{\mathcal J}_a&= \boldsymbol{\mathcal J}_{a0}^{\rm rel}+\boldsymbol{X}_a^{(0)},
&
\boldsymbol{\mathcal G}_{ab} &=  \frac{1}{c}\boldsymbol{\mathcal J}_{ab}^{\rm rel},
&
\boldsymbol{\mathcal S}_a&=  \frac{1}{2c^{2}}\left(\boldsymbol{\mathcal J}_{a0}^{\rm rel}-\boldsymbol{X}_a^{(0)}\right),\\
\boldsymbol{\mathcal H}_a&= \boldsymbol{\mathcal P}_{a0}^{\rm rel}+\boldsymbol{X}_a^{(1)},
&
\boldsymbol{\mathcal P}_{ab} &=  \frac{1}{c}\boldsymbol{\mathcal P}_{ab}^{\rm rel},
&
\boldsymbol{\mathcal M}_a&=  \frac{1}{2c^{2}}\left(\boldsymbol{\mathcal P}_{a0}^{\rm rel}-\boldsymbol{X}_a^{(1)}\right).
\end{aligned}
\end{equation}
Similarly, using \eqref{redefsl3iiexp} one gets \eqref{IWpoin1} and
\begin{equation}\label{IWpoin3}
\begin{aligned}
\boldsymbol{\mathcal J}_{ab}&= \boldsymbol{\mathcal J}_{ab}^{\rm rel}+\boldsymbol{X}_{ab}^{(0)},
&
\boldsymbol{\mathcal G}_a &=  \frac{1}{c}\boldsymbol{\mathcal J}_{a0}^{\rm rel},
&
\boldsymbol{\mathcal S}_{ab}&=  \frac{1}{2c^{2}}\left(\boldsymbol{\mathcal J}_{ab}^{\rm rel}-\boldsymbol{X}_{ab}^{(0)}\right),\\
\boldsymbol{\mathcal H}_{ab}&= \boldsymbol{\mathcal P}_{ab}^{\rm rel}+\boldsymbol{X}_{ab}^{(1)},
&
\boldsymbol{\mathcal P}_a &=  \frac{1}{c}\boldsymbol{\mathcal P}_{a0}^{\rm rel},
&
\boldsymbol{\mathcal M}_{ab}&=  \frac{1}{2c^{2}}\left(\boldsymbol{\mathcal P}_{ab}^{\rm rel}-\boldsymbol{X}_{ab}^{(1)}\right).
\end{aligned}
\end{equation}
Thus, putting it all together one finds that $\mathfrak{hs_3 barg1_{2+1}}$ can be obtained from the contraction defined by \eqref{IWpoin1} and \eqref{IWpoin2}, while $\mathfrak{hs_3 barg2_{2+1}}$ follows from taking $c\rightarrow$ in \eqref{IWpoin1} and \eqref{IWpoin3}. Schematically, the NR limis \eqref{expreli} and \eqref{exprelii} for $S=S^{(1)}_E$ can be written as
\begin{equation}\label{contractionshspoin}
\begin{aligned}
&\mathfrak{hs_3 poin_{2+1}}\oplus \mathfrak{iso}(3)\;&
\underset{c\rightarrow\infty}{\longrightarrow}& \qquad \mathfrak{hs_3 barg1_{2+1}}
\\[6pt]
&\mathfrak{hs_3 poin_{2+1}} \oplus \mathfrak{isl}(2,\mathbb R)\oplus \mathfrak{so}(2)\oplus\mathfrak{so}(2)\;&\underset{c\rightarrow\infty}{\longrightarrow}&\qquad \mathfrak{hs_3 barg2_{2+1}}
\end{aligned}
\end{equation}
Moreover, one can see from \eqref{IWpoin1} that dropping the higher-spin generators precisely corresponds to the contraction relating the Poincar\'e and the Extended Bargmann algebras in three space-time dimensions \cite{Bergshoeff:2016lwr}. In other words, when removing the spin-$3$ generators, the left-hand sides of the contractions in \eqref{contractionshspoin} boil down to $\mathfrak{iso}(2,1)\oplus \mathfrak{so}(2) \oplus\mathfrak{so}(2)$ whereas the right-hand sides reduce to the Extended Bargman algebra in $2+1$ dimensions.

\subsection{Spin-$3$ non-relativistic Maxwell algebras}
\label{NRMax}
The Maxwell algebra is an extension of the Poincar\'e algebra that describes particle systems in the presence of a constant electromagnetic field \cite{Bacry:1970ye,Schrader:1972zd}. In $2+1$ dimensions, the electric NR limit of the Maxwell algebra \cite{Gonzalez:2016xwo,Rubio:2018itx,Gomis:2019fdh} admits three central extensions \cite{Aviles:2018jzw} (there dubbed \emph{exotic}), in complete analogy with the two central extensions appearing in the Extended Bargmann algebra. This particular extended NR limit\footnote{Other NR limits of the Maxwell algebra have been considered in \cite{Bacry:1970du,Beckers:1983gp,negro1990local2,Gomis:2019fdh}.}, which from now on in this paper will be simply referred to as NR Maxwell algebra in $2+1$ dimensions, is isomorphic to the expanded Nappi-Witten algebra $S_E^{(2)}\times\mathfrak{nw}$ \cite{Penafiel:2019czp}, and its commutation relations are \eqref{ExtBargAlg} plus
\begin{equation}\label{ExtNRMaxAlg}
\begin{aligned}
\left[\boldsymbol{J},\boldsymbol{Z}_a\right]=\left[\boldsymbol{H},\boldsymbol{P}_a\right]=
\left[\boldsymbol{Y},\boldsymbol{G}_a\right]=\epsilon_a^{\;\;b}\boldsymbol{Z}_b,\qquad
\left[\boldsymbol{G}_a,\boldsymbol{Z}_b\right]=\left[\boldsymbol{P}_a,\boldsymbol{P}_b\right]=-\epsilon_{ab}\boldsymbol{T},
\end{aligned}
\end{equation}
where the expanded Nappi-Witten generators are the ones given in \eqref{BargRenameGen} and
\begin{equation}\label{MaxwellRenameGen}
\boldsymbol{Y}=\boldsymbol{k}^{(2)},\quad \boldsymbol{Z}_a=\boldsymbol{k}_b^{(2)}, \quad \boldsymbol{T}=\boldsymbol{\sigma}^{(2)}.
\end{equation}
In the NR Maxwell algebra, the generators $\boldsymbol{Y}$ and $\boldsymbol{Z}_a$ are associated to the magnetic and the electric field that define the constant electromagnetic background, while $\boldsymbol{T}$ is a third central term (which, as in the previous case, ceases being central when the higher-spin generators are introduced). One can define two inequivalent spin-$3$ extensions of the NR Maxwell algebra in three dimensions by evaluating the expanded algebras \eqref{HSNWalgebra1exp} and \eqref{HSNWalgebra2exp} for $S=S^{(2)}_E$, i.e.
\begin{equation}\label{hsmiandii}
\mathfrak{hs_3 nr\textnormal{-}max1_{2+1}}=S^{(2)}_E\times \mathfrak{hs_3 nw1},\qquad
\mathfrak{hs_3 nr\textnormal{-}max2_{2+1}}=S^{(2)}_E\times \mathfrak{hs_3 nw2}.
\end{equation}
\begin{itemize}
\item In order to define the first higher-spin extension $\mathfrak{hs_3 nr\textnormal{-}max1_{2+1}}$, we evaluate the commutation relations of $S^{(2)}_E\times\mathfrak{hs_3 nw1}$ using \eqref{HSNWalgebra1exp} and the semigroup product law \eqref{sen} for $n=2$. We define
\begin{equation}\label{hsMaxwell1RenameGen}
\boldsymbol{\mathcal Y}_a=\boldsymbol{\mathcal K}_a^{(2)},\quad \boldsymbol{\mathcal Z}_{ab}=\boldsymbol{\mathcal K}_{ab}^{(2)},\quad \boldsymbol{\mathcal T}_a=\boldsymbol{\Sigma}_a^{(2)},
\end{equation}
which, together with the set of expanded generators appearing in Eqs. \eqref{BargRenameGen}, \eqref{hsBarg1RenameGen} and \eqref{MaxwellRenameGen}, satisfy the commutation relations \eqref{ExtBargAlg}, \eqref{BargHS1}, \eqref{BargHS2} \eqref{ExtNRMaxAlg} and
\begin{equation}\label{MaxHS}
\begin{aligned}
&\left[\boldsymbol{J},\boldsymbol{\mathcal Y}_a\right]=\left[\boldsymbol{Y},\boldsymbol{\mathcal J}_a\right] = \left[\boldsymbol{H},\boldsymbol{\mathcal H}_a\right] = \epsilon_{ab}\boldsymbol{\mathcal Y}_b,\\[5pt]
&\left[\boldsymbol{J},\boldsymbol{\mathcal Z}_{ab}\right]=\left[\boldsymbol{Y},\boldsymbol{\mathcal G}_{ab}\right]  =  \left[\boldsymbol{H},\boldsymbol{\mathcal P}_{ab}\right]  = -\epsilon_{c(a}\boldsymbol{\mathcal Z}_{b)c},\\[5pt]
&\left[\boldsymbol{\mathcal J}_a,\boldsymbol{\mathcal Y}_b\right]  =  \left[\boldsymbol{\mathcal H}_a,\boldsymbol{\mathcal H}_b\right]  =\epsilon_{ab}\boldsymbol{Y},\\[5pt]
&\left[\boldsymbol{\mathcal J}_a, \boldsymbol{\mathcal T}_b\right] =\left[\boldsymbol{\mathcal Y}_a, \boldsymbol{\mathcal S}_b\right]=\left[\boldsymbol{\mathcal H}_a, \boldsymbol{\mathcal M}_b\right]= \epsilon_{ab}\boldsymbol{T},\\[5pt]
&\left[\boldsymbol{\mathcal G}_{ab},\boldsymbol{\mathcal Z}_{cd}\right] = \left[\boldsymbol{\mathcal P}_{ab},\boldsymbol{\mathcal P}_{cd}\right] =\delta_{(a(c}\epsilon_{d)b)}\boldsymbol{T},\\[5pt]
&\left[\boldsymbol{G}_a,\boldsymbol{\mathcal Y}_b\right] = \left[\boldsymbol{Z}_a,\boldsymbol{\mathcal J}_b\right]  = \left[\boldsymbol{P}_a,\boldsymbol{\mathcal H}_b\right] =-\epsilon_{a(c}\boldsymbol{\mathcal Z}_{b)c},
\\[5pt]
&\left[\boldsymbol{\mathcal J}_a,\boldsymbol{\mathcal Z}_{bc}\right]  =\left[\boldsymbol{\mathcal Y}_a,\boldsymbol{\mathcal G}_{bc}\right]   =\left[\boldsymbol{\mathcal H}_a,\boldsymbol{\mathcal P}_{bc}\right]  =\delta_{a(b}\epsilon_{c)d}\boldsymbol{Z}_d,
\\[5pt]
&\left[\boldsymbol{G}_a,\boldsymbol{\mathcal Z}_{bc}\right]  =  \left[\boldsymbol{Z}_a,\boldsymbol{\mathcal G}_{bc}\right]  = \left[\boldsymbol{P}_a,\boldsymbol{\mathcal P}_{bc}\right]  =-\epsilon_{a(b}\boldsymbol{\mathcal T}_{c)},
\\[5pt]
&\left[\boldsymbol{J},\boldsymbol{\mathcal T}_a\right] =\left[\boldsymbol{S},\boldsymbol{\mathcal Y}_a\right]  = \left[\boldsymbol{Y},\boldsymbol{\mathcal S}_a\right]=\left[\boldsymbol{T},\boldsymbol{\mathcal J}_a\right]  =\left[\boldsymbol{H},\boldsymbol{\mathcal M}_a\right]=\left[\boldsymbol{M},\boldsymbol{\mathcal H}_a\right]    =\epsilon_{ab}\boldsymbol{\mathcal T}_b.
\\[5pt]
\end{aligned}
\end{equation}
\item The second spin-$3$ extension of the NR Maxwell symmetry, $\mathfrak{hs_3 nr\textnormal{-}max2_{2+1}}$, follows from defining
\begin{equation}\label{hsMaxwell2RenameGen}
\boldsymbol{\mathcal Y}_{ab}=\tilde{\boldsymbol{\mathcal K}}_{ab}^{(2)},\quad \boldsymbol{\mathcal Z}_a=\tilde{\boldsymbol{\mathcal K}}_a^{(2)},\quad \boldsymbol{\mathcal T}_{ab}=\tilde{\boldsymbol{\Sigma}}_{ab}^{(2)},
\end{equation}
in addition to \eqref{BargRenameGen}, \eqref{hsBarg2RenameGen} and \eqref{MaxwellRenameGen}. Using again the multiplication law of the semigroup $S^{(2)}_E$ the algebra $\mathfrak{hs_3 nr\textnormal{-}max2_{2+1}}$ can be read off from \eqref{HSNWalgebra2exp} and is given by \eqref{ExtBargAlg}, \eqref{BargHS1T}, \eqref{BargHS2T}, \eqref{ExtNRMaxAlg} together with the commutation realations
\begin{equation}\label{MaxHST}
\begin{aligned}
&[\boldsymbol{J},\boldsymbol{\mathcal Z}_a ]=[\boldsymbol{Y},\boldsymbol{\mathcal G}_a ] =[\boldsymbol{H},\boldsymbol{\mathcal P}_a ] =\epsilon_{ab}\boldsymbol{\mathcal Z}_b,\\[5pt]
&[\boldsymbol{J},\boldsymbol{\mathcal Y}_{ab}] =[\boldsymbol{Y},\boldsymbol{\mathcal J}_{ab}]=[\boldsymbol{H},\boldsymbol{\mathcal H}_{ab}]  =-\epsilon_{c(a}\boldsymbol{\mathcal Y}_{b)c},\\[5pt]
&[\boldsymbol{\mathcal J}_{ab},\boldsymbol{\mathcal Y}_{cd}]=[\boldsymbol{\mathcal H}_{ab},\boldsymbol{\mathcal H}_{cd}]  =\delta_{(a(c}\epsilon_{d)b)} \boldsymbol{Y},\\[5pt]
&[\boldsymbol{\mathcal G}_a ,\boldsymbol{\mathcal Z}_b ]=[\boldsymbol{\mathcal P}_a ,\boldsymbol{\mathcal P}_b ]  =\epsilon_{ab}\boldsymbol{T},\\[5pt]
&[\boldsymbol{G}_a ,\boldsymbol{\mathcal Z}_b ]=[\boldsymbol{Z}_a ,\boldsymbol{\mathcal G}_b ]=[\boldsymbol{P}_a ,\boldsymbol{\mathcal P}_b ]=-\epsilon_{a(c}\boldsymbol{\mathcal T}_{b)c},\\[5pt]
&[\boldsymbol{G}_a ,\boldsymbol{\mathcal Y}_{bc}]=[\boldsymbol{Z}_a ,\boldsymbol{\mathcal J}_{bc}]=[\boldsymbol{P}_a ,\boldsymbol{\mathcal H}_{bc}]=-\epsilon_{a(b}\boldsymbol{\mathcal Z}_{c)},
\\[5pt]
&[\boldsymbol{\mathcal G}_a ,\boldsymbol{\mathcal Y}_{bc}] =[\boldsymbol{\mathcal Z}_a ,\boldsymbol{\mathcal J}_{bc}] =[\boldsymbol{\mathcal P}_a ,\boldsymbol{\mathcal H}_{bc}] =\delta_{a(b}\epsilon_{c)d}\boldsymbol{Z}_d,
\\[5pt]
&[\boldsymbol{\mathcal J}_{ab},\boldsymbol{\mathcal T}_{cd}] =[\boldsymbol{\mathcal Y}_{ab},\boldsymbol{\mathcal S}_{cd}] =[\boldsymbol{\mathcal H}_{ab},\boldsymbol{\mathcal M}_{cd}] =\delta_{(a(c}\epsilon_{d)b)} \boldsymbol{T},
\\[5pt]
&[\boldsymbol{J},\boldsymbol{\mathcal T}_{ab}] =[\boldsymbol{Y},\boldsymbol{\mathcal S}_{ab}] =[\boldsymbol{S},\boldsymbol{\mathcal Y}_{ab}]=[\boldsymbol{T},\boldsymbol{\mathcal J}_{ab}] =[\boldsymbol{H},\boldsymbol{\mathcal M}_{ab}] =[\boldsymbol{M},\boldsymbol{\mathcal H}_{ab}] =-\epsilon_{c(a}\boldsymbol{\mathcal T}_{b)c}.
\end{aligned}
\end{equation}
\end{itemize}
In the same way as in the Bargmann case, one can rename the higher-spin generators in the algebras given above either as
\begin{equation}\label{swap2}
\begin{aligned}
&\boldsymbol{\mathcal H}_a\leftrightarrow \boldsymbol{\mathcal Y}_a,&\quad&
\boldsymbol{\mathcal P}_{ab}\leftrightarrow \boldsymbol{\mathcal Z}_{ab},&\quad&
\boldsymbol{\mathcal M}_a\leftrightarrow \boldsymbol{\mathcal T}_a,\\
&\boldsymbol{\mathcal H}_{ab}\leftrightarrow \boldsymbol{\mathcal Y}_{ab},&\quad&
\boldsymbol{\mathcal P}_a\leftrightarrow \boldsymbol{\mathcal Z}_b,&\quad&
\boldsymbol{\mathcal M}_{ab}\leftrightarrow \boldsymbol{\mathcal T}_{ab},
\end{aligned}
\end{equation}
or as
\begin{equation}\label{swap3}
\begin{aligned}
&\boldsymbol{\mathcal J}_a\leftrightarrow \boldsymbol{\mathcal Y}_a,&\quad&
\boldsymbol{\mathcal G}_{ab}\leftrightarrow \boldsymbol{\mathcal Z}_{ab},&\quad&
\boldsymbol{\mathcal S}_a\leftrightarrow \boldsymbol{\mathcal T}_a,\\
&\boldsymbol{\mathcal J}_{ab}\leftrightarrow \boldsymbol{\mathcal Y}_{ab},&\quad&
\boldsymbol{\mathcal G}_a\leftrightarrow \boldsymbol{\mathcal Z}_b,&\quad&
\boldsymbol{\mathcal S}_{ab}\leftrightarrow \boldsymbol{\mathcal T}_{ab},
\end{aligned}
\end{equation}
giving isomorphic spin-$3$ extensions of the NR Maxwell algebra in $2+1$ dimensions, which can be physically different depending on how the higher-spin generators are physically interpreted. Similarly one can try the combinations \eqref{swap1}, \eqref{swap2} and \eqref{swap1}, \eqref{swap3}.

It is interesting to note that the expanded algebras $S^{(2)}_E\times\mathfrak{hs_3 nw1}$ and $S^{(2)}_E\times\mathfrak{hs_3 nw2}$ also define spin-$3$ extensions of a NR version of the Hietarinta algebra \cite{Bansal:2018qyz,Chernyavsky:2019hyp} in $2+1$ dimensions if one interchanges the generators
\begin{equation}
\boldsymbol{H}\leftrightarrow \boldsymbol{Y},\quad
\boldsymbol{P}_a\leftrightarrow \boldsymbol{Z}_a,\quad
\boldsymbol{M}\leftrightarrow \boldsymbol{T},\\
\end{equation}
in the NR Maxwell sublagebra defined by Eqs. \eqref{ExtBargAlg} and \eqref{ExtNRMaxAlg}.
\\

\noindent\textbf{Extended semidirect sum structure}: The relativistic Maxwell algebra has the structure of an extended semidirect sum, where the Poincar\'e translations become non-Abelian. On the other hand, the higher-spin symmetry $\mathfrak{hs_3 nr\textnormal{-}max1_{2+1}}$ ($\mathfrak{hs_3 nr\textnormal{-}max2_{2+1}}$) extends the algebra $\mathfrak{hs_3 barg1_{2+1}}$ ($\mathfrak{hs_3 barg2_{2+1}}$) found in Section \ref{HSB} by rendering its Abelian ideal non-Abelian. Since the extra generators in this extension form, in turn, a new Abelian ideal that can be identified with $\mathfrak{hs_3 nw1}^{({\rm ab})}$ ($\mathfrak{hs_3 nw1}^{({\rm ab})}$), one can conclude that $\mathfrak{hs_3 nr\textnormal{-}max1_{2+1}}$ ($\mathfrak{hs_3 nr\textnormal{-}max2_{2+1}}$) is an extended semidirect sum algebra of the same type as the original Maxwell algebra in three space-time dimensions. Indeed, the commutation relations of the $\mathfrak{hs_3 nr\textnormal{-}max1_{2+1}}$ ($\mathfrak{hs_3 nr\textnormal{-}max2_{2+1}}$) algebra can be written in a compact form as \cite{Salgado-Rebolledo:2019kft}
\begin{equation}
\left[(X,X^\ast,X^\prime),(Y,Y^\ast,Y^\prime)\right]=\left([X,Y],[X,Y^\ast]-[Y,X^\ast],[X,Y^\prime]-[Y,X^\prime]+[X^\ast,Y^\ast]\right),
\end{equation}
with $X$ an element of $\mathfrak{hs_3 nw1}$ ($\mathfrak{hs_3 nw2}$), and $X^\ast$ and $X^\prime$ belonging to $\mathfrak{hs_3 nw1}^{({\rm ab})}$ ($\mathfrak{hs_3 nw2}^{({\rm ab})}$). Therefore, the spin-$3$ extensions of the Extended Bargmann algebra in $2+1$ dimensions can be expressed as
\begin{equation}\label{extsds}
\begin{aligned}
\mathfrak{hs_3 nr\textnormal{-}max1_{2+1}}& = \mathfrak{hs_3 nw1} \oright _{\rm ad}  \mathfrak{hs_3 nw1}^{({\rm ab})}_{\rm ext},\\
\mathfrak{hs_3 nr\textnormal{-}max2_{2+1}}&= \mathfrak{hs_3 nw2} \oright _{\rm ad}  \mathfrak{hs_3 nw2}^{({\rm ab})}_{\rm ext},
\end{aligned}
\end{equation}
where $\mathfrak{hs_3 nw1}^{({\rm ab})}_{\rm ext}$ ($\mathfrak{hs_3 nw2}^{({\rm ab})}_{\rm ext}$) denotes the extended group $\mathfrak{hs_3 nw1} \otimes\mathfrak{hs_3 nw1}$ ($\mathfrak{hs_3 nw2} \otimes\mathfrak{hs_3 nw2}$) endowed with the product
\begin{equation}\label{exttranslations}
\left(X^\ast,X^\prime\right)\hat{+}\left(Y^\ast,Y^\prime \right)=\left(X^\ast+Y^\ast\,,\; X^\prime+Y^\prime +\frac{1}{2}\,[X^\ast,Y^\ast] \right).
\end{equation}
The relativistic analog of \eqref{extsds} is the extended semidirect sum that defines the spin-$3$ extension of the Maxwell algebra\footnote{A higher-spin extensions of the Maxwell algebra in higher dimensions has been considered in \cite{Fedoruk:2013sna}.} in $2+1$ dimensions \cite{Salgado-Rebolledo:2019kft}
\begin{equation}\label{hsmax}
\mathfrak{hs_3 max_{2+1}}\simeq\mathfrak{sl}(3,\mathbb R)  \oright _{\rm ad}  \mathfrak{sl}(3,\mathbb R)^{({\rm ab})}_{\rm ext}.
\end{equation}

\noindent\textbf{NR limit}: As shown in the general case treated in Section \ref{NRlim}, the higher-spin NR symmetries \eqref{hsmiandii} can be obtained as NR limits applied to relativistic symmetries. The algebra \eqref{hsmax} was constructed in \cite{Caroca:2017izc} as the expansion
\begin{equation}\label{hsmax2}
\mathfrak{hs_3 max_{2+1}}\simeq S^{(2)}_E\times \mathfrak{sl}(3,\mathbb R),
\end{equation}
with the expanded generators \eqref{hspoingen} plus
\begin{equation}\label{hsmaxgen}
\boldsymbol{Z}_A^{\rm rel} = \boldsymbol{L}^{(2)}_A,\quad
\boldsymbol{\mathcal Z}_{AB}^{\rm rel} = \boldsymbol{\mathcal L}^{(2)}_{AB},
\end{equation}
and commutation relations \eqref{sl3exp} for $S=S^{(2)}_E$. Now we look at the expanded algebras \eqref{so3exp} and \eqref{sl2xu1exp}, necessary to define the NR limits \eqref{redefsl3iexp} and \eqref{redefsl3iiexp}. When using the semigroup product law of $S^{(2)}_E$ given in \eqref{sen}, these can be shown to be isomorphic to the Euclidean Maxwell algebra in three dimensions $\mathfrak{e\textnormal{-}max_{3}}$ (spanned by $\{\boldsymbol{X}^{(0)},\boldsymbol{X}^{(1)},\boldsymbol{X}_a^{(0)},\boldsymbol{X}_a^{(1)},\boldsymbol{X}_a^{(2)}\}$ in Eq. \eqref{so3exp}) and of the direct sum of the Maxwell algebra in $2+1$ dimensions and three Abelian generators (spanned by $\{\boldsymbol{X}^{(0)},\boldsymbol{X}^{(1)},\boldsymbol{X}_{ab}^{(0)},\boldsymbol{X}_{ab}^{(1)},\boldsymbol{X}_{ab}^{(2)}\}$ in Eq. \eqref{sl2xu1exp}),
\begin{equation}
\begin{aligned}
&S^{(2)}_E\times \mathfrak{so}(3) \simeq \mathfrak{e\textnormal{-}max_{3}},\\
&S^{(2)}_E\times\Big( \mathfrak{sl}(2,\mathbb R)\oplus \mathfrak{so}(2)\Big) \simeq \mathfrak{max_{2+1}}\oplus \mathfrak{so}(2)\oplus\mathfrak{so}(2)\oplus\mathfrak{so}(2).
\end{aligned}
\end{equation}
Using the relation \eqref{redefsl3iexp} to relate relativistic and NR generators, we find for the NR Maxwell generators the relations \eqref{IWpoin1} plus
\begin{equation}\label{IWmax1}
\begin{aligned}
\boldsymbol{Y} & =  \boldsymbol{Z}_{0}^{\rm rel}+\boldsymbol{X}^{(2)}, &
\boldsymbol{Z}_a & =  \frac{1}{c}\boldsymbol{Z}_a^{\rm rel}, &
\boldsymbol{T} & =  \frac{1}{2c^2}\left(\boldsymbol{Z}_{0}^{\rm rel}-\boldsymbol{X}^{(2)}\right),
\end{aligned}
\end{equation}
whereas the higher-spin generators of $\mathfrak{hs_3 nr\textnormal{-}max1_{2+1}}$ satisfy \eqref{IWpoin2} and
\begin{equation}\label{IWmax2}
\begin{aligned}
\boldsymbol{\mathcal Y}_a&= \boldsymbol{\mathcal Z}_{a0}^{\rm rel}+\boldsymbol{X}_a^{(2)},
&
\boldsymbol{\mathcal Z}_{ab} &=  \frac{1}{c}\boldsymbol{\mathcal Z}_{ab}^{\rm rel},
&
\boldsymbol{\mathcal T}_a&=  \frac{1}{2c^{2}}\left(\boldsymbol{\mathcal Z}_{a0}^{\rm rel}-\boldsymbol{X}_a^{(2)}\right).\\
\end{aligned}
\end{equation}
The second possible contraction \eqref{redefsl3iiexp}, yields once again \eqref{IWpoin1} and \eqref{IWmax1}, but the higher-spin extension is given by \eqref{IWpoin3} together with
\begin{equation}\label{IWmax3}
\begin{aligned}
\boldsymbol{\mathcal Y}_{ab}&= \boldsymbol{\mathcal Z}_{ab}^{\rm rel}+\boldsymbol{X}_{ab}^{(2)},
&
\boldsymbol{\mathcal Z}_a &=  \frac{1}{c}\boldsymbol{\mathcal Z}_{a0}^{\rm rel},
&
\boldsymbol{\mathcal T}_{ab}&=  \frac{1}{2c^{2}}\left(\boldsymbol{\mathcal Z}_{ab}^{\rm rel}-\boldsymbol{X}_{ab}^{(2)}\right),
\end{aligned}
\end{equation}
and leads to $\mathfrak{hs_3 nr\textnormal{-}max2_{2+1}}$ in the limit $c\rightarrow\infty$. Therefore, the spin-$3$ algebras \eqref{hsmiandii} follow from the NR limits
\begin{equation}\label{contractionshsmax}
\begin{aligned}
&\mathfrak{hs_3 max_{2+1}}\oplus \mathfrak{e\textnormal{-}max_{3}}\;&
\underset{c\rightarrow\infty}{\longrightarrow}& \qquad \mathfrak{hs_3 nr\textnormal{-}max1_{2+1}}
\\[6pt]
&\mathfrak{hs_3 max_{2+1}} \oplus \mathfrak{max_{2+1}}\oplus \mathfrak{so}(2)\oplus\mathfrak{so}(2)\oplus\mathfrak{so}(2)\;&\underset{c\rightarrow\infty}{\longrightarrow}&\qquad \mathfrak{hs_3 nr\textnormal{-}max2_{2+1}}
\end{aligned}
\end{equation}
The relations \eqref{IWpoin1} and \eqref{IWmax1} show that, in absence of spin-$3$ generators, the standard contraction of the Maxwell algebra in three space-time dimensions and three Abelian generators that yields the NR Maxwell algebra \cite{Aviles:2018jzw} is recovered.

\subsection{Spin-$3$ Newton-Hooke algebras}

One can include a cosmological constant in the Bargman algebra by considering a NR contraction of the (anti-)deSitter algebra \cite{Bacry:1968zf}. This leads to the Newton-Hooke algebra, which in three space-time dimensions admits two central extensions \cite{Hartong:2016yrf} and is given by the extended Bargamann commutation relations \eqref{ExtBargAlg} plus
\begin{equation}\label{generatorsnh}
\left[\boldsymbol{H},\boldsymbol{P}_a\right]= \epsilon_{ab}\boldsymbol{G}_b,\quad
\left[\boldsymbol{P}_a,\boldsymbol{P}_b\right]=- \epsilon_{ab}\;\boldsymbol{S}.
\end{equation}
The Extended Newton-Hooke algebra is isomorphic to the expanded Nappi-Witten algebra $S_M ^{(1)}\times \mathfrak{nw}$. Thus, we can construct the corresponding spin-$3$ extensions by evaluating expanded algebras
\begin{equation}\label{hsniandii}
\mathfrak{hs_3 nh1_{2+1}}=S^{(1)}_M \times \mathfrak{hs_3 nw1},\qquad
\mathfrak{hs_3 nh2_{2+1}} = S^{(1)}_M \times \mathfrak{hs_3 nw1}
\end{equation}
We find the explicit form of their commutation relations by using the product law of the semigroup $S^{(1)}_M\backsimeq\mathbb{Z}_2$ given in \eqref{smn} for $n=1$.
\begin{itemize}
\item We define the first higher-spin extension of the Extended Newton-Hooke algebra in $2+1$ dimensions algebra, $\mathfrak{hs_3 nh1_{2+1}}$, by defining the higher spin generators \eqref{hsBarg1RenameGen}, which  yields the commutation relations \eqref{ExtBargAlg}, \eqref{BargHS1}, \eqref{BargHS2}, \eqref{generatorsnh} and
\begin{equation}\label{NewtonHookeHS}
\begin{aligned}
&\left[\boldsymbol{H},\boldsymbol{\mathcal H}_a\right]  =  \epsilon_{ab}\boldsymbol{\mathcal J}_b,&
&\left[\boldsymbol{H},\boldsymbol{\mathcal P}_{ab}\right] =  -  \epsilon_{c(a}\boldsymbol{\mathcal G}_{b)c},&
&\left[\boldsymbol{H},\boldsymbol{\mathcal M}_a\right] =\left[\boldsymbol{M},\boldsymbol{\mathcal H}_a\right]  =  \epsilon_{ab}\boldsymbol{\mathcal S}_b,
\\[5pt]
&\left[\boldsymbol{P}_a,\boldsymbol{\mathcal H}_b\right] = -  \epsilon_{a(c}\boldsymbol{\mathcal G}_{b)c},&
&\left[\boldsymbol{P}_a,\boldsymbol{\mathcal P}_{bc}\right]  =  -  \epsilon_{a(b}\boldsymbol{\mathcal S}_{c)}, &
&\left[\boldsymbol{\mathcal H}_a,\boldsymbol{\mathcal H}_b\right]  =   \epsilon_{ab}\boldsymbol{J},
\\[5pt]
&\left[\boldsymbol{\mathcal H}_a, \boldsymbol{\mathcal M}_b\right] =  \epsilon_{ab}\boldsymbol{S},&
&\left[\boldsymbol{\mathcal H}_a,\boldsymbol{\mathcal P}_{bc}\right]  = \delta_{a(b}\epsilon_{c)d}\boldsymbol{G}_d,&
&\left[\boldsymbol{\mathcal P}_{ab},\boldsymbol{\mathcal P}_{cd}\right] =  \delta_{(a(c}\epsilon_{d)b)}\boldsymbol{S}.
\end{aligned}
\end{equation}
\item The commutation relations of the second higher-spin extension $\mathfrak{hs_3 nh2_{2+1}}$ are obtained by considering the expanded generators \eqref{hsBarg2RenameGen}. The result is the relations \eqref{ExtBargAlg}, \eqref{BargHS1T}, \eqref{BargHS2T}, \eqref{generatorsnh} plus
\begin{equation}\label{NewtonHookeHST}
\begin{aligned}
&[\boldsymbol{H},\boldsymbol{\mathcal P}_a ] =\epsilon_{ab}\boldsymbol{\mathcal G}_b,&
&[\boldsymbol{H},\boldsymbol{\mathcal H}_{ab}] =-\epsilon_{c(a}\boldsymbol{\mathcal J}_{b)c},&
&[\boldsymbol{H},\boldsymbol{\mathcal M}_{ab}] =[\boldsymbol{M},\boldsymbol{\mathcal H}_{ab}] =-\epsilon_{c(a}\boldsymbol{\mathcal S}_{b)c},
\\[5pt]
&[\boldsymbol{P}_a ,\boldsymbol{\mathcal P}_b ]=-\epsilon_{a(c}\boldsymbol{\mathcal S}_{b)c},&
&[\boldsymbol{P}_a ,\boldsymbol{\mathcal H}_{bc}]=-\epsilon_{a(b}\boldsymbol{\mathcal G}_{c)},&
&[\boldsymbol{\mathcal H}_{ab},\boldsymbol{\mathcal H}_{cd}] =\delta_{(a(c}\epsilon_{d)b)} \boldsymbol{J}.
\\[5pt]
&[\boldsymbol{\mathcal P}_a ,\boldsymbol{\mathcal P}_b ] =\epsilon_{ab}\boldsymbol{S},&
&[\boldsymbol{\mathcal P}_a ,\boldsymbol{\mathcal H}_{bc}] =\delta_{a(b}\epsilon_{c)d}\boldsymbol{G}_d,&
&[\boldsymbol{\mathcal H}_{ab},\boldsymbol{\mathcal M}_{cd}] =\delta_{(a(c}\epsilon_{d)b)} \boldsymbol{S}.
\end{aligned}
\end{equation}
\end{itemize}

\noindent\textbf{Direct sum structure}: The Extended Newton-Hooke algebra defined by \eqref{ExtBargAlg} and \eqref{generatorsnh} is isomorphic to a direct product of two copies of the Nappi-Witten algebra \cite{Alvarez:2007ys,Hartong:2017bwq,Joung:2018frr}, which is a NR analog of the isomorphism $\mathfrak{so}(2,2)\backsimeq\mathfrak{sl}(2,\mathbb R)\oplus\mathfrak{sl}(2,\mathbb R)$. Similarly, in the spin-$3$ case, the following isomorphisms hold
\begin{equation}\begin{aligned}\label{hsnhiso}
&\mathfrak{hs_3 nh1_{2+1}} \backsimeq \mathfrak{hs_3 nw1}\oplus \mathfrak{hs_3 nw1},\\
&\mathfrak{hs_3 nh2_{2+1}} \backsimeq \mathfrak{hs_3 nw2}\oplus \mathfrak{hs_3 nw2}.
\end{aligned}
\end{equation}
Indeed, the combinations
\begin{equation}\label{2nw}
\boldsymbol{k}^{\pm}=\frac{1}{2}\left( \boldsymbol{J} \pm  \boldsymbol{H} \right),\quad
\boldsymbol{k}^{\pm}_a=\frac{1}{2}\left( \boldsymbol{G}_a \pm  \boldsymbol{P}_a \right),\quad
\boldsymbol{\sigma}^{\pm}=\frac{1}{2}\left( \boldsymbol{S} \pm \boldsymbol{M} \right),
\end{equation}
satisfy the $\mathfrak{nw}$ commutation relations \eqref{nwalg}. The isomorphism \eqref{hsnhiso} can be easily proven by using change of basis \eqref{2nw} and the higher-spin generalization
\begin{equation}\label{2hsnw1}
\boldsymbol{\mathcal K}^{\pm}_a=\frac{1}{2}\left( \boldsymbol{\mathcal J}_a \pm  \boldsymbol{\mathcal H}_a \right),\quad
\boldsymbol{\mathcal K}^{\pm}_{ab}=\frac{1}{2}\left( \boldsymbol{\mathcal G}_{ab} \pm  \boldsymbol{\mathcal P}_{ab} \right),\quad
\boldsymbol{\Sigma}^{\pm}_a=\frac{1}{2}\left( \boldsymbol{\mathcal S}_a \pm \boldsymbol{\mathcal M}_a \right),
\end{equation}
in the $\mathfrak{hs_3 nh1_{2+1}}$ algebra. Instead, supplementing \eqref{nwalg} with
\begin{equation}\label{2hsnw2}
\tilde{\boldsymbol{\mathcal K}}^{\pm}_{ab}=\frac{1}{2}\left( \boldsymbol{\mathcal J}_{ab} \pm  \boldsymbol{\mathcal H}_{ab} \right),\quad
\tilde{\boldsymbol{\mathcal K}}^{\pm}_a=\frac{1}{2}\left( \boldsymbol{\mathcal G}_a \pm  \boldsymbol{\mathcal P}_a\right),\quad
\tilde{\boldsymbol{\Sigma}}^{\pm}_{ab}=\frac{1}{2}\left( \boldsymbol{\mathcal S}_{ab} \pm \boldsymbol{\mathcal M}_{ab} \right),
\end{equation}
does the job for the $\mathfrak{hs_3 nh2_{2+1}}$ algebra. It is important to remark that all the previous result consider a negative cosmological constant, (the Newton-Hooke symmetry in our analysis is stricktly speaking Newton-Hooke$^-$). The results can be generalized to the case of positive cosmological constant by redefining the generators
\begin{equation}\label{dscase1}
\begin{aligned}
&\boldsymbol{H}\rightarrow i\boldsymbol{H},&\quad &\boldsymbol{P}_a\rightarrow i\boldsymbol{P}_a,&\quad&\boldsymbol{M}\rightarrow i\boldsymbol{M}\\
&\boldsymbol{\mathcal H}_a\rightarrow i\boldsymbol{\mathcal H}_a,&\quad &\boldsymbol{\mathcal P}_{ab}\rightarrow i\boldsymbol{\mathcal P}_{ab},&\quad&\boldsymbol{\mathcal M}_a\rightarrow i\boldsymbol{\mathcal M}_a\\
&\boldsymbol{\mathcal H}_{ab}\rightarrow i\boldsymbol{\mathcal H}_{ab},&\quad &\boldsymbol{\mathcal P}_a\rightarrow i\boldsymbol{\mathcal P}_a,&\quad&\boldsymbol{\mathcal M}_{ab}\rightarrow i\boldsymbol{\mathcal M}_{ab}.
\end{aligned}
\end{equation}

\noindent\textbf{NR and flat limits}: As in the Bargmann case, we can obtain \eqref{hsniandii} by taking a NR limit of suitable extensions of the spin-$3$ AdS algebra \cite{Campoleoni:2010zq}
\begin{equation}\label{hsadsiso}
\mathfrak{hs_3 ads_{2+1}}\simeq \mathfrak{sl}(3,\mathbb R)\oplus\mathfrak{sl}(3,\mathbb R).
\end{equation}
Indeed, it is clear that one can obtain \eqref{hsnhiso} from \eqref{hsadsiso} using two copies of the contractions developed in Section \ref{HSNW} that take $\mathfrak{sl}(3,\mathbb R)$ into the NR algebras $\mathfrak{hs_3 nw1}$ and $\mathfrak{hs_3 nw2}$. Moreover, \eqref{hsadsiso} is equivalent to the expansion
\begin{equation}\label{hsadsiso2}
\mathfrak{hs_3 ads_{2+1}}\simeq S^{(1)}_M\times \mathfrak{sl}(3,\mathbb R),
\end{equation}
introduced in \cite{Caroca:2017izc}, where the commutation relations can be read off from \eqref{sl3exp} using the product law of the semigroup $S^{(1)}_M$ and expanded generators \eqref{hspoingen} defined in the Poincar\'e case. Thus, one can define the expanded algebras \eqref{expreli} and \eqref{exprelii} for $S=S^{(1)}_M$ and take the NR limit to find \eqref{hsnhiso}. This requires to extend \eqref{hsadsiso2} with the algebras \eqref{so3exp} and \eqref{sl2xu1exp}, which we can identify using the following isomorphisms (see \eqref{diagexpso3}, \eqref{diagexpsltw} and \eqref{expuo}):
\begin{equation}\label{otherexpsads}
\begin{aligned}
&S^{(1)}_M\times \mathfrak{so}(3)  \simeq \mathfrak{so}(3)\oplus\mathfrak{so}(3)  \simeq\mathfrak{so}(4),\\
&S^{(1)}_M\times\Big( \mathfrak{sl}(2,\mathbb R)\oplus \mathfrak{so}(2)\Big) \simeq \mathfrak{so}(2,2)\oplus \mathfrak{so}(2)\oplus\mathfrak{so}(2).
\end{aligned}
\end{equation}

As shown in Section \ref{flatlimit}, one can define a vanishing cosmological constant limit that takes \eqref{hsniandii} into \eqref{hsbiandii} by means of the rescaling \eqref{recalingsell}, i.e.
\eqref{BargRenameGen}, \eqref{hsBarg1RenameGen} and \eqref{hsBarg2RenameGen}
\begin{equation}\label{rescaleP's}
\begin{aligned}
&\boldsymbol{H}\rightarrow\ell\,\boldsymbol{H},& &\boldsymbol{P}_a\rightarrow\ell\,\boldsymbol{P}_a, & &\boldsymbol{M}\rightarrow\ell\,\boldsymbol{M},\\
&\boldsymbol{\mathcal H}_a\rightarrow\ell\,\boldsymbol{\mathcal H}_a,&
&\boldsymbol{\mathcal P}_{ab}\rightarrow\ell\,\boldsymbol{\mathcal P}_{ab},&
&\boldsymbol{\mathcal M}_a\rightarrow\ell\,\boldsymbol{\mathcal M}_a,\\
&\boldsymbol{\mathcal H}_{ab}\rightarrow\ell\,\boldsymbol{\mathcal H}_{ab},&
&\boldsymbol{\mathcal P}_a\rightarrow\ell\,\boldsymbol{\mathcal P}_a,& & \boldsymbol{\mathcal M}_{ab}\rightarrow\ell\,\boldsymbol{\mathcal M}_{ab},
\end{aligned}
\end{equation}
whose effect is to add a factor $1/\ell^2$ on the right-hand side of the commutation relations given in Eqs. \eqref{generatorsnh}, \eqref{NewtonHookeHS} and \eqref{NewtonHookeHST}, so that they dissapear in the limit $\ell\rightarrow\infty$ yielding the contractions. Similarly, the contraction defined by \eqref{recalingsellRel}, \eqref{rescellX1} and \eqref{rescellX2} in this case are
\begin{equation}\begin{aligned}\label{reslcaleP'sRel}
&\boldsymbol{P}_A^{\rm rel}\rightarrow \ell\,\boldsymbol{P}_A^{\rm rel},&\quad&
\boldsymbol{\mathcal P}_{AB}^{\rm rel}\rightarrow \ell\,\boldsymbol{\mathcal P}_{AB}^{\rm rel}\\
&\boldsymbol{X}^{(1)}\rightarrow \ell\,\boldsymbol{X}^{(1)},&\quad&
\boldsymbol{X}_a^{(1)}\rightarrow \ell\,\boldsymbol{X}_a^{(1)},\quad
\boldsymbol{X}_{ab}^{(1)}\rightarrow \ell\,\boldsymbol{X}_{ab}^{(1)}
\end{aligned}\end{equation}
and define the analog contraction at the relativistic level, taking \eqref{hsadsiso2} and \eqref{otherexpspoin} into \eqref{hspoin2} and \eqref{otherexpsads}, respectively. Thus, the NR contraction leading to $\mathfrak{hs_3 nh1_{2+1}}$ together with the corresponding flat limit is
\begin{equation}\label{fulldiagram1ads}
\begin{aligned}
&\mathfrak{hs_3 ads_{2+1}}\oplus \mathfrak{so}(4)
\;
&\underset{c\rightarrow\infty}{\longrightarrow}& \qquad \mathfrak{hs_3 nh1_{2+1}}
\\
&\hskip1.0truecm {\scriptstyle \ell\rightarrow\infty}\;\Big\downarrow  & &\hskip.9truecm {\scriptstyle \ell\rightarrow\infty}\;\Big\downarrow\\
&\mathfrak{hs_3 poin_{2+1}}\oplus \mathfrak{iso}(3)
\;
&\underset{c\rightarrow\infty}{\longrightarrow}& \qquad \mathfrak{hs_3 barg1_{2+1}}
\end{aligned}
\end{equation}
whereas the corresponding diagram for $\mathfrak{hs_3 nh2_{2+1}}$ has the form
\begin{equation}\label{fulldiagram2ads}
\begin{aligned}
&\mathfrak{hs_3 ads_{2+1}} \oplus \mathfrak{so}(2,2)\oplus\mathfrak{so}(2)\oplus\mathfrak{so}(2)
\;
&\underset{c\rightarrow\infty}{\longrightarrow}& \qquad \mathfrak{hs_3 nh2_{2+1}}
\\
&\hskip2.3truecm {\scriptstyle \ell\rightarrow\infty}\;\Big\downarrow  & &\hskip.9truecm {\scriptstyle \ell\rightarrow\infty}\;\Big\downarrow\\
&\mathfrak{hs_3 poin_{2+1}} \oplus \mathfrak{isl}(2,\mathbb R)\oplus \mathfrak{so}(2)\oplus\mathfrak{so}(2)
\;
&\underset{c\rightarrow\infty}{\longrightarrow}& \qquad \mathfrak{hs_3 barg2_{2+1}}
\end{aligned}
\end{equation}
In both cases, the explicit relations of the $c\rightarrow\infty$ contractions are the same as the ones given in the Poincar\'e case \eqref{IWpoin1}-\eqref{IWpoin3}.

\subsection{Spin-$3$ non-relativistic AdS-Lorentz algebras}
As the last example, we turn our attention to the AdS-Lorentz symmetry. Based on the fact that NR AdS-Lorentz algebra is the expanded Nappi-Witten algebra \eqref{expNW} for $S=S^{(2)}_M$, we define the corresponding spin-$3$ extensions as
\begin{equation}\label{hsaliandii}
\mathfrak{hs_3 nr\textnormal{-}adslor1_{2+1}}= S^{(2)}_M\times \mathfrak{hs_3 nw1}, \qquad
\mathfrak{hs_3 nr\textnormal{-}adslor2_{2+1}}= S^{(2)}_M \times \mathfrak{hs_3 nw1}.
\end{equation}
The relativistic AdS-Lorentz algebra a generalization of the Maxwell algebra that provides a semi-simple extension of the Poincar\'e algebra \cite{Soroka:2006aj,Gomis:2009dm}, and whose NR limit has been considered in the context of higher-dimensional (extended) CS theories \cite{Gonzalez:2016xwo,Rubio:2018itx}. In $2+1$dimensions it admits three central extensions and its commutation relations are the ones given in \eqref{ExtBargAlg} and \eqref{ExtNRMaxAlg}, plus
\begin{equation}\label{nonreladslalg}
\begin{aligned}
&\left[\boldsymbol{Y},\boldsymbol{Z}_a\right]= \epsilon_{a}^{\;\;b} \boldsymbol{Z}_b\,,&\quad&
\left[\boldsymbol{P}_a,\boldsymbol{Z}_b\right]=- \epsilon_{ab} \boldsymbol{M},\\[5pt]
&\left[\boldsymbol{H},\boldsymbol{Z}_a\right]= \left[\boldsymbol{Y},\boldsymbol{P}_a\right]= \epsilon_{a}^{\;\;b}\boldsymbol{P}_b,&\quad&
 \left[\boldsymbol{Z}_a,\boldsymbol{Z}_b\right]=- \epsilon_{ab}\boldsymbol{T},
\end{aligned}
\end{equation}
As said before, this algebra follows from evaluating the expanded Nappi-Witten algebra \eqref{expNW} for the semigroup $S^{n}_M$ defined in \eqref{smn}, and setting $n=2$. The same procedure allows one to find the commutation relations of the higher-spin extensions \eqref{hsaliandii}
\begin{itemize}
\item The first higher-spin generalization of the NR AdS-Lorentz algebra $\mathfrak{hs_3 nr\textnormal{-}adslor1_{2+1}}$  includes the higher spin generators \eqref{hsBarg1RenameGen} and \eqref{hsMaxwell1RenameGen}. In this case the commutation relations are the ones of the $\mathfrak{hs_3 nr\textnormal{-}max1_{2+1}}$ algebra,  \eqref{ExtBargAlg}, \eqref{BargHS1}, \eqref{BargHS2}, \eqref{ExtNRMaxAlg}, \eqref{MaxHS}, plus \eqref{nonreladslalg} and
\begin{equation}\label{AdSLHS1}
\begin{aligned}
&\left[\boldsymbol{Y},\boldsymbol{\mathcal Y}_a\right]  =   \epsilon_{ab}\boldsymbol{\mathcal Y}_b,
&
&\left[\boldsymbol{Y},\boldsymbol{\mathcal Z}_{ab}\right] =  -  \epsilon_{c(a}\boldsymbol{\mathcal Z}_{b)c},&
&\left[\boldsymbol{Y},\boldsymbol{\mathcal T}_a\right] =\left[\boldsymbol{T},\boldsymbol{\mathcal Y}_a\right]  =   \epsilon_{ab}\boldsymbol{\mathcal T}_b,
\\[5pt]
&\left[\boldsymbol{Z}_a,\boldsymbol{\mathcal Y}_b\right] = -  \epsilon_{a(c}\boldsymbol{\mathcal Z}_{b)c},
&
&\left[\boldsymbol{Z}_a,\boldsymbol{\mathcal Z}_{bc}\right]  =  -  \epsilon_{a(b}\boldsymbol{\mathcal T}_{c)}, &
&\left[\boldsymbol{\mathcal Y}_a,\boldsymbol{\mathcal Y}_b\right]  =    \epsilon_{ab}\boldsymbol{Y},
\\[5pt]
&\left[\boldsymbol{\mathcal Y}_a, \boldsymbol{\mathcal T}_b\right] =   \epsilon_{ab}\boldsymbol{T},&
&\left[\boldsymbol{\mathcal Y}_a,\boldsymbol{\mathcal Z}_{bc}\right]  =  \delta_{a(b}\epsilon_{c)d}\boldsymbol{Z}_d,&
&\left[\boldsymbol{\mathcal Z}_{ab},\boldsymbol{\mathcal Z}_{cd}\right] =   \delta_{(a(c}\epsilon_{d)b)}\boldsymbol{T},
\end{aligned}
\end{equation}
and
\begin{equation}\label{AdSLHS2}
\begin{aligned}
&\left[\boldsymbol{Y},\boldsymbol{\mathcal H}_a\right]=\left[\boldsymbol{H},\boldsymbol{\mathcal Y}_a\right]   = \epsilon_{ab}\boldsymbol{\mathcal H}_b,&
&\left[\boldsymbol{Z}_a,\boldsymbol{\mathcal H}_b\right] = \left[\boldsymbol{P}_a,\boldsymbol{\mathcal Y}_b\right] = -\epsilon_{a(c}\boldsymbol{\mathcal P}_{b)c},
\\[5pt]
&\left[\boldsymbol{Y},\boldsymbol{\mathcal P}_{ab}\right]=\left[\boldsymbol{H},\boldsymbol{\mathcal Z}_{ab}\right]  =  -\epsilon_{c(a}\boldsymbol{\mathcal P}_{b)c},&
&\left[\boldsymbol{\mathcal Y}_a,\boldsymbol{\mathcal P}_{bc}\right]  =\left[\boldsymbol{\mathcal H}_a,\boldsymbol{\mathcal Z}_{bc}\right]   =\delta_{a(b}\epsilon_{c)d}\boldsymbol{P}_d,
\\[5pt]
&\left[\boldsymbol{\mathcal Y}_a,\boldsymbol{\mathcal H}_b\right]  =  \epsilon_{ab}\boldsymbol{H},&
&\left[\boldsymbol{Z}_a,\boldsymbol{\mathcal P}_{bc}\right]  =  \left[\boldsymbol{P}_a,\boldsymbol{\mathcal Z}_{bc}\right]  =-\epsilon_{a(b}\boldsymbol{\mathcal M}_{c)},
\\[5pt]
&\left[\boldsymbol{\mathcal Y}_a, \boldsymbol{\mathcal M}_b\right] =\left[\boldsymbol{\mathcal H}_a, \boldsymbol{\mathcal T}_b\right]= \epsilon_{ab}\boldsymbol{M},&
&\left[\boldsymbol{Y},\boldsymbol{\mathcal M}_a\right] =\left[\boldsymbol{T},\boldsymbol{\mathcal H}_a\right]  = \left[\boldsymbol{H},\boldsymbol{\mathcal T}_a\right]=\left[\boldsymbol{M},\boldsymbol{\mathcal Y}_a\right]  =\epsilon_{ab}\boldsymbol{\mathcal M}_b.
\\[5pt]
&\left[\boldsymbol{\mathcal Z}_{ab},\boldsymbol{\mathcal P}_{cd}\right] = \delta_{(a(c}\epsilon_{d)b)}\boldsymbol{M},
\end{aligned}
\end{equation}

\item The second higher-spin extension of the NR AdS-Lorentz algebra $\mathfrak{hs_3 nr\textnormal{-}adslor2_{2+1}}$  includes the higher spin generators \eqref{hsBarg2RenameGen} and \eqref{hsMaxwell2RenameGen} and has the commutation relations of the $\mathfrak{hs_3 nr\textnormal{-}max2_{2+1}}$ algebra,  \eqref{ExtBargAlg}, \eqref{BargHS1T}, \eqref{BargHS2T}, \eqref{ExtNRMaxAlg}, \eqref{MaxHST}, plus \eqref{nonreladslalg} and
\begin{equation}\label{AdSLHS1T}
\begin{aligned}
&[\boldsymbol{Y},\boldsymbol{\mathcal Z}_a ] =\epsilon_{ab}\boldsymbol{\mathcal Z}_b,&
&[\boldsymbol{Y},\boldsymbol{\mathcal Y}_{ab}] =-\epsilon_{c(a}\boldsymbol{\mathcal Y}_{b)c},&
&[\boldsymbol{Y},\boldsymbol{\mathcal T}_{ab}] =[\boldsymbol{T},\boldsymbol{\mathcal Y}_{ab}] =-\epsilon_{c(a}\boldsymbol{\mathcal T}_{b)c},
\\[5pt]
&[\boldsymbol{Z}_a ,\boldsymbol{\mathcal Z}_b ]=-\epsilon_{a(c}\boldsymbol{\mathcal T}_{b)c},&
&[\boldsymbol{Z}_a ,\boldsymbol{\mathcal Y}_{bc}]=-\epsilon_{a(b}\boldsymbol{\mathcal Z}_{c)},&
&[\boldsymbol{\mathcal Y}_{ab},\boldsymbol{\mathcal Y}_{cd}] =\delta_{(a(c}\epsilon_{d)b)} \boldsymbol{Y}.
\\[5pt]
&[\boldsymbol{\mathcal Z}_a ,\boldsymbol{\mathcal Z}_b ] =\epsilon_{ab}\boldsymbol{T},&
&[\boldsymbol{\mathcal Z}_a ,\boldsymbol{\mathcal Y}_{bc}] =\delta_{a(b}\epsilon_{c)d}\boldsymbol{Z}_d,&
&[\boldsymbol{\mathcal Y}_{ab},\boldsymbol{\mathcal T}_{cd}] =\delta_{(a(c}\epsilon_{d)b)} \boldsymbol{T},
\end{aligned}
\end{equation}
plus
\begin{equation}\label{AdSLHS2T}
\begin{aligned}
&[\boldsymbol{Y},\boldsymbol{\mathcal P}_a ]=[\boldsymbol{H},\boldsymbol{\mathcal Z}_a ] =\epsilon_{ab}\boldsymbol{\mathcal P}_b,&
&[\boldsymbol{Z}_a ,\boldsymbol{\mathcal H}_{bc}]=[\boldsymbol{P}_a ,\boldsymbol{\mathcal Y}_{bc}]=-\epsilon_{a(b}\boldsymbol{\mathcal P}_{c)},
\\[5pt]
&[\boldsymbol{Y},\boldsymbol{\mathcal H}_{ab}] =[\boldsymbol{H},\boldsymbol{\mathcal Y}_{ab}] =-\epsilon_{c(a}\boldsymbol{\mathcal H}_{b)c},&
&[\boldsymbol{\mathcal Z}_a ,\boldsymbol{\mathcal H}_{bc}] =[\boldsymbol{\mathcal P}_a ,\boldsymbol{\mathcal Y}_{bc}] =\delta_{a(b}\epsilon_{c)d}\boldsymbol{P}_d,
\\[5pt]
&[\boldsymbol{\mathcal Y}_{ab},\boldsymbol{\mathcal H}_{cd}] =\delta_{(a(c}\epsilon_{d)b)} \boldsymbol{H},&
&[\boldsymbol{\mathcal Y}_{ab},\boldsymbol{\mathcal M}_{cd}] =[\boldsymbol{\mathcal H}_{ab},\boldsymbol{\mathcal T}_{cd}] =\delta_{(a(c}\epsilon_{d)b)} \boldsymbol{M},
\\[5pt]
&[\boldsymbol{\mathcal Z}_a ,\boldsymbol{\mathcal P}_b ] =\epsilon_{ab}\boldsymbol{M},&
&[\boldsymbol{Y},\boldsymbol{\mathcal M}_{ab}] =[\boldsymbol{H},\boldsymbol{\mathcal T}_{ab}] =[\boldsymbol{T},\boldsymbol{\mathcal H}_{ab}]
\\[5pt]
&[\boldsymbol{Z}_a ,\boldsymbol{\mathcal P}_b ]=[\boldsymbol{P}_a ,\boldsymbol{\mathcal Z}_b ]=-\epsilon_{a(c}\boldsymbol{\mathcal M}_{b)c},&
&\hskip1.7truecm =[\boldsymbol{M},\boldsymbol{\mathcal Y}_{ab}] =-\epsilon_{c(a}\boldsymbol{\mathcal M}_{b)c}.
\end{aligned}
\end{equation}
\end{itemize}

\noindent\textbf{Direct sum structure}: The AdS-Lorentz symmetry in $d+1$ dimensions was so dubbed because it is isomorphic to the direct $\mathfrak{so}(d,2)\oplus\mathfrak{so}(d,1)$ \cite{Gomis:2009dm}. Along the same lines, the NR AdS Lorentz algebra is isomorphic to the direct sum of the Extended Newton-Hooke algebra and the Nappi-Witten algebra, which in turn is equivalent to three copies of the $\mathfrak{nw}$ algebra. The spin-$3$ generalization of this statement reads
\begin{equation}\begin{aligned}
&\mathfrak{hs_3 nr\textnormal{-}adslor1_{2+1}} \backsimeq \mathfrak{hs_3 nh1_{2+1}}\oplus \mathfrak{hs_3 nw1} \backsimeq \mathfrak{hs_3 nw1}\oplus \mathfrak{hs_3 nw1}\oplus \mathfrak{hs_3 nw1},\\
&\mathfrak{hs_3 nr\textnormal{-}adslor2_{2+1}} \backsimeq \mathfrak{hs_3 nh2_{2+1}}\oplus \mathfrak{hs_3 nw2} \backsimeq \mathfrak{hs_3 nw2}\oplus \mathfrak{hs_3 nw2}\oplus \mathfrak{hs_3 nw2},
\end{aligned}
\end{equation}
where, on the right-hand side of this relation, we have used \eqref{hsnhiso} (This is the NR analog of the fact that, in $2+1$  dimensions, the AdS Lorentz algebra is isomorphi to the direct sum of three copies of $\mathfrak{sl}(2,\mathbb R)$ \cite{Concha:2018jjj}). This can be shown by considering the change of basis
\begin{align}
& \hat{\boldsymbol{J}}= \boldsymbol{Y} ,&\quad&\hat{\boldsymbol{G}}_a= \boldsymbol{Z}_a,&\quad&\hat{\boldsymbol{S}}= \boldsymbol{T},\nonumber \\
& \hat{\boldsymbol{\mathcal J}}_a= \boldsymbol{\mathcal Y}_a ,&\quad&\hat{\boldsymbol{\mathcal G}}_{ab}= \boldsymbol{\mathcal Z}_{ab},&\quad&\hat{\boldsymbol{\mathcal S}}_a= \boldsymbol{\mathcal T}_a, \label{redefadsl1}\\
& \hat{\boldsymbol{\mathcal J}}_{ab}= \boldsymbol{\mathcal Y}_{ab} ,&\quad&\hat{\boldsymbol{\mathcal G}}_a= \boldsymbol{\mathcal Z}_a,&\quad&\hat{\boldsymbol{\mathcal S}}_{ab}= \boldsymbol{\mathcal T}_{ab},\nonumber
\\[9pt]
& \hat{\boldsymbol{H}}= \boldsymbol{H} ,&\quad&\hat{\boldsymbol{P}}_a= \boldsymbol{P}_a,&\quad& \hat{\boldsymbol{M}}= \boldsymbol{M},\nonumber \\
& \hat{\boldsymbol{\mathcal H}}_a= \boldsymbol{\mathcal H}_a ,&\quad&\hat{\boldsymbol{\mathcal P}}_{ab}= \boldsymbol{\mathcal P}_{ab},&\quad& \hat{\boldsymbol{\mathcal M}}_a= \boldsymbol{\mathcal M}_a, \label{redefadsl2}\\
& \hat{\boldsymbol{\mathcal H}}_{ab}= \boldsymbol{\mathcal H}_{ab} ,&\quad&\hat{\boldsymbol{\mathcal P}}_a= \boldsymbol{\mathcal P}_a,&\quad& \hat{\boldsymbol{\mathcal M}}_{ab}= \boldsymbol{\mathcal M}_{ab},\nonumber
\\[9pt]
&  \hat{\boldsymbol{k}}= \boldsymbol{J} - \boldsymbol{Y}  ,&\quad&\hat{\boldsymbol{k}}_a= \boldsymbol{G}_a- \boldsymbol{Z}_a ,&\quad&   \hat{\boldsymbol{\sigma}}=\boldsymbol{S}- \boldsymbol{T}, \nonumber \\
&  \hat{\boldsymbol{\mathcal K}}_a= \boldsymbol{\mathcal J}_a - \boldsymbol{\mathcal Y}_a  ,&\quad&\hat{\boldsymbol{\mathcal K}}_{ab}= \boldsymbol{\mathcal G}_{ab}- \boldsymbol{\mathcal Z}_{ab} ,&\quad&   \hat{\boldsymbol{\Sigma}}_a=\boldsymbol{\mathcal S}_a- \boldsymbol{\mathcal T}_a, \label{redefadsl3} \\
&  \hat{\tilde{\boldsymbol{\mathcal K}}}_{ab}= \boldsymbol{\mathcal J}_{ab} - \boldsymbol{\mathcal Y}_{ab}  ,&\quad&\hat{\tilde{\boldsymbol{\mathcal K}}}_a= \boldsymbol{\mathcal G}_a- \boldsymbol{\mathcal Z}_a ,&\quad&   \hat{\tilde{\boldsymbol{\Sigma}}}_{ab}=\boldsymbol{\mathcal S}_{ab}- \boldsymbol{\mathcal T}_{ab}.  \nonumber
\end{align}
The first and second lines in Eqs. \eqref{redefadsl1}-\eqref{redefadsl3} take the $\mathfrak{hs_3 nr\textnormal{-}adslor1_{2+1}}$ algebra  into the direct sum of $\mathfrak{hs_3 nh1_{2+1}}$ and $\mathfrak{hs_3 nw1}$, whereas the first and third lines in Eqs. \eqref{redefadsl1}-\eqref{redefadsl3} do the analog job for $\mathfrak{hs_3 nr\textnormal{-}adslor2_{2+1}}$. Clearly, one can apply the change of basis \eqref{2nw}-\eqref{2hsnw2} to the hatted generators to end up with three copies of a spin-$3$ Nappi-Witten algebra in each case. The previous results can be extended to the case of a positive cosmological constant by using the redefintion \eqref{dscase1} together with
\begin{equation}
\begin{aligned}
&\boldsymbol{Y}\rightarrow  -\boldsymbol{Y},&\quad &\boldsymbol{Z}_a\rightarrow  -\boldsymbol{Z}_a,&\quad&\boldsymbol{T}\rightarrow-  \boldsymbol{T}\\
&\boldsymbol{\mathcal Y}_a\rightarrow -\boldsymbol{\mathcal Y}_a,&\quad &\boldsymbol{\mathcal Z}_{ab}\rightarrow -\boldsymbol{\mathcal Z}_{ab},&\quad&\boldsymbol{\mathcal T}_a\rightarrow -\boldsymbol{T}_a\\
&\boldsymbol{\mathcal Y}_{ab}\rightarrow -\boldsymbol{\mathcal Y}_{ab},&\quad &\boldsymbol{\mathcal Z}_a\rightarrow -\boldsymbol{\mathcal Z}_a,&\quad&\boldsymbol{\mathcal T}_{ab}\rightarrow -\boldsymbol{T}_{ab}.
\end{aligned}
\end{equation}

\noindent\textbf{NR and flat limits}: We can construct relativistic symmetries that reproduce \eqref{hsaliandii} in the $c\rightarrow\infty$ starting from the higher-spin extension of the AdS-Lorentz symmetry in $2+1$ dimensions \cite{Caroca:2017izc}
\begin{equation}\label{hsadsL}
\mathfrak{hs_3 adslor_{2+1}}=S^{(2)}_M\times \mathfrak{sl}(3,\mathbb R).
\end{equation}
The commutation relations of \eqref{hsadsL} can be read off from Eq. \eqref{sl3exp} with the same definition \eqref{hspoingen} and \eqref{hsmaxgen} for the expanded generators as the Maxwellian case. We extend this symmetry to define the algebras \eqref{expreli} and \eqref{expreli} for $S=S^{(2)}_M$, which requires introducing
\begin{equation}\label{otherexpsadslor}
\begin{aligned}
&S^{(2)}_M\times \mathfrak{so}(3) \simeq \mathfrak{e\textnormal{-}adslor_{3}}\simeq \mathfrak{so}(3)\oplus\mathfrak{so}(3)\oplus\mathfrak{so}(3),\\
&S^{(2)}_M\times\Big( \mathfrak{sl}(2,\mathbb R)\oplus \mathfrak{so}(2)\Big) \simeq \mathfrak{adslor_{2+1}}\oplus \mathfrak{so}(2)\oplus\mathfrak{so}(2)\oplus\mathfrak{so}(2).
\end{aligned}
\end{equation}
The NR contractions have the same form as the one found in the Maxwell case, which follows from the general expressions \eqref{redefsl3iexp} and \eqref{redefsl3iiexp}, and are explicitely given by \eqref{IWpoin1}, \eqref{IWpoin2}, \eqref{IWmax1}, \eqref{IWmax2}, and \eqref{IWpoin1}, \eqref{IWpoin3}, \eqref{IWmax1}, \eqref{IWmax3}, respectively.

The previous results can be connected with the ones found in Section \ref{NRMax} for the Maxwell case by means of a contraction analog to the vanishing cosmological limit that relates the Extended Newton-Hooke and the Bargmann symmetry. The flat limit in this case extends the relations \eqref{rescaleP's} by introducing the extra rescalings
\begin{equation}\label{rescaleZ's}
\begin{aligned}
&\boldsymbol{Y}\rightarrow\ell^2\,\boldsymbol{Y},& &\boldsymbol{Z}_a\rightarrow\ell^2\,\boldsymbol{Z}_a, & &\boldsymbol{T}\rightarrow\ell^2\,\boldsymbol{T},\\
&\boldsymbol{\mathcal Y}_a\rightarrow\ell^2\,\boldsymbol{\mathcal Y}_a,&
&\boldsymbol{\mathcal Z}_{ab}\rightarrow\ell^2\,\boldsymbol{\mathcal Z}_{ab},&
&\boldsymbol{\mathcal T}_a\rightarrow\ell^2\,\boldsymbol{\mathcal T}_a,\\
&\boldsymbol{\mathcal Y}_{ab}\rightarrow\ell^2\,\boldsymbol{\mathcal Y}_{ab},&
&\boldsymbol{\mathcal Z}_a\rightarrow\ell^2\,\boldsymbol{\mathcal Z}_a,&
& \boldsymbol{\mathcal T}_{ab}\rightarrow\ell^2\,\boldsymbol{\mathcal T}_{ab},
\end{aligned}
\end{equation}
as defined in \eqref{recalingsell}, which sets all the commutators given in Eqs. \eqref{nonreladslalg}-\eqref{AdSLHS2T} to zero in the limit $\ell^2\rightarrow\infty$. For the relativistic generators we use the rescaling \eqref{recalingsellRel}, \eqref{rescellX1} and \eqref{rescellX2}, which lead to \eqref{reslcaleP'sRel} plus
\begin{equation}\begin{aligned}\label{reslcaleZ'sRel}
&\boldsymbol{Z}_A^{\rm rel}\rightarrow \ell^2\,\boldsymbol{Z}_A^{\rm rel},&\quad&
\boldsymbol{\mathcal Z}_{AB}^{\rm rel}\rightarrow \ell^2\,\boldsymbol{\mathcal Z}_{AB}^{\rm rel},\\
&\boldsymbol{X}^{(2)}\rightarrow \ell^2\,\boldsymbol{X}^{(2)},&\quad&
\boldsymbol{X}_a^{(2)}\rightarrow \ell^2\,\boldsymbol{X}_a^{(2)},\quad
\boldsymbol{X}_{ab}^{(2)}\rightarrow \ell^2\,\boldsymbol{X}_{ab}^{(2)}.
\end{aligned}\end{equation}
Thus, in the case of the $\mathfrak{hs_3 nr\textnormal{-}adslor1_{2+1}}$ algebra we find the following diagram analog to \eqref{fulldiagram1ads}\begin{equation}\label{fulldiagram1adslor}
\begin{aligned}
&\mathfrak{hs_3 adslor_{2+1}}\oplus\mathfrak{so}(3)\oplus\mathfrak{so}(3)\oplus\mathfrak{so}(3)
\;
&\underset{c\rightarrow\infty}{\longrightarrow}& \qquad \mathfrak{hs_3 nr\textnormal{-}adslor1_{2+1}}
\\
&\hskip1.5truecm {\scriptstyle \ell\rightarrow\infty}\;\Big\downarrow  & &\hskip.9truecm {\scriptstyle \ell\rightarrow\infty}\;\Big\downarrow\\
&\mathfrak{hs_3 max_{2+1}}\oplus \mathfrak{e\textnormal{-}max_{3}}
\;
&\underset{c\rightarrow\infty}{\longrightarrow}& \qquad \mathfrak{hs_3 nr\textnormal{-}max1_{2+1}}
\end{aligned}
\end{equation}
while in the case of $\mathfrak{hs_3 nr\textnormal{-}adslor2_{2+1}}$ the corresponding diagram, analog to \eqref{fulldiagram2ads}, has the form
\begin{equation}\label{fulldiagram2adslor}
\begin{aligned}
&\mathfrak{hs_3 adslor_{2+1}} \oplus \mathfrak{adslor_{2+1}}\oplus \mathfrak{so}(2)\oplus\mathfrak{so}(2)\oplus\mathfrak{so}(2)
\;
&\underset{c\rightarrow\infty}{\longrightarrow}& \qquad \mathfrak{hs_3 nr\textnormal{-}adslor2_{2+1}}
\\
&\hskip2.3truecm {\scriptstyle \ell\rightarrow\infty}\;\Big\downarrow  & &\hskip.9truecm {\scriptstyle \ell\rightarrow\infty}\;\Big\downarrow\\
&\mathfrak{hs_3 max_{2+1}} \oplus \mathfrak{max_{2+1}}\oplus \mathfrak{so}(2)\oplus\mathfrak{so}(2)\oplus\mathfrak{so}(2)
\;
&\underset{c\rightarrow\infty}{\longrightarrow}& \qquad \mathfrak{hs_3 nr\textnormal{-}max2_{2+1}}
\end{aligned}
\end{equation}
In both cases, the explicit relations of the $c\rightarrow\infty$ contractions are the same as the ones given in the Poincar\'e case \eqref{IWpoin1}-\eqref{IWpoin3}.

\subsection{Higher-spin gravity theories}
\label{HSG}

Higher-spin gravity actions invariant under the spin-$3$ symmetries presented above can be constructed using the defintions \eqref{expAs}, \eqref{expAs} in \eqref{CSform}, together with the invariant tensors defined by \eqref{itnwexp}, \eqref{itexp1} and \eqref{itnwexp}, \eqref{itexp2}, which correspond to the expansions $S\times\mathfrak{hs_3 nw1}$ and $S\times\mathfrak{hs_3 nw2}$, respectively. A considerable easier task is to use the results \eqref{hsCS1} and \eqref{hsCS2} and simply evaluate the semigroup product law $i\diamond j$ for the semigroups under consideration. Here, adopt the latter strategy, and describe the steps to be followed in order to find the result. Since the form of the resulting CS forms turns out to be rather cumbersome, the explicit form of the Lagrangians and the details of the derivations are given in Appendix \ref{appA}.

\begin{itemize}
\item \textbf{Spin-$3$ Bargmann algebras}: the action \eqref{CSNWexp} for $S=S_E^{(1)}$ is the one of Extended Bargmann gravity \cite{Papageorgiou:2009zc,Bergshoeff:2016lwr,Hartong:2016yrf,Penafiel:2016ufo} with gauge symmetry \eqref{ExtBargAlg}, and thus, the following gauge fields are identified
\begin{equation}\label{changeFieldsb}
\begin{aligned}
 \alpha_{(0)}= w, &\quad&&\theta_{(0)}^a= \omega^a, &\quad&
\beta_{(0)} =s,\\
\alpha_{(1)}= \tau, &\quad& &\theta_{(1)}^a= e^a, &\quad&\beta_{(1)}= m,\\
\end{aligned}
\end{equation}
where $w$ and $\omega^a$ are the spin connections associated with rotations and Galilean boosts, respectively, $\tau$ is the clock form, $e^a$ is the spatial dreibein, and $m$ and $s$ are the gauge fields that go along the mass and the spin generators.
The higher-spin extensions \eqref{hsCS1} and \eqref{hsCS2} introduce a spin-$3$ version of each generators given in Eq. \eqref{changeFieldsb}. It is therefore natural to adopt the convention
\begin{equation}\label{changeFieldsb1}
\begin{aligned}
& A^a_{(0)}= W^a,&\quad&\Theta_{(0)}^{ab}= \Omega^{ab},&\quad&
B^a_{(0)} =S^a,\\
& A^a_{(1)}= V^a,&\quad&\Theta_{(1)}^{ab}= E^{ab},&\quad& B^a_{(1)}= M^a,
\end{aligned}
\end{equation}
when defining the gauge fields associated to the $\mathfrak{hs_3 barg1_{2+1}}$ algebra. Similarly, when considering the $\mathfrak{hs_3 barg2_{2+1}}$ symmetry, one can define the spin-$3$ extension of \eqref{changeFieldsb} as
\begin{equation}\label{changeFieldsb2}
\begin{aligned}
&A^{ab}_{(0)}= W^{ab},&\quad &\Theta_{(0)}^a= \Omega^a,&\quad&
B^{ab}_{(0)} =S^{ab},\\
&A^{ab}_{(1)}= V^{ab},&\quad&\Theta_{(1)}^a= E^a, &\quad& B^{ab}_{(1)}= M^{ab}.
\end{aligned}
\end{equation}
The explicit form of the spin-$3$ CS gravity Lagrangians \eqref{hsCS1} and \eqref{hsCS2} for $S^{(1)}_E$ follows from Eqs. \eqref{hsCSNH1} and \eqref{hsCSNH2} of Appendix \ref{appNHB} after taking the limit $\ell\rightarrow\infty$.

\item \textbf{Spin-$3$ Maxwell algebras}: considering $S^{(2)}_E$ in the CS form \eqref{CSNWexp} yields defines a NR Maxwellian CS gravity theory \cite{Aviles:2018jzw,Penafiel:2019czp}, for the gravitational gauge fields \eqref{changeFieldsb} and the Maxwellian gauge fields
\begin{equation}\label{changeFieldsm}
\alpha_{(2)}= y,\quad \theta_{(2)}^a= k^a,\quad\beta_{(2)}= t.
\end{equation}
One can then extend such theory by evaluating the Lagrangian \eqref{hsCS2} for $S^{(2)}_E$, which yields a CS action with $\mathfrak{hs_3 nr\textnormal{-}max1_{2+1}}$ symmetry gauge symmetry that includes the spin-$3$ gauge fields \eqref{changeFieldsb2} plus
\begin{equation}\label{changeFieldsm1}
A^a_{(2)}= Y^a,\quad \Theta_{(2)}^{ab}= K^{ab},\quad B^a_{(2)}= T^a.
\end{equation}
Similarly, the $\mathfrak{hs_3 nr\textnormal{-}max2_{2+1}}$-invariant CS theory coming from \eqref{hsCS2}, in terms of \eqref{changeFieldsb2} and
\begin{equation}\label{changeFieldsm2}
A^{ab}_{(2)}= Y^{ab},\quad \Theta_{(2)}^a= K^a,\quad B^{ab}_{(2)}= T^{ab}.
\end{equation}
These are respectively given in Eqs. \eqref{hsCSNRAdSL1} and \eqref{hsCSNRAdSL2} of Appendix \ref{appAdSLM} when the limit $\ell\rightarrow\infty$ is taken.
\item \textbf{Spin-$3$ Newton-Hooke algebras}: replacing $S=S^{(1)}_M$ in the CS form \eqref{CSNWexp} yields Extended Newton-Hooke CS gravity \cite{Papageorgiou:2010ud,Hartong:2016yrf,Penafiel:2016ufo}.
When considering the extensions $\mathfrak{hs_3 nh1_{2+1}}$ and $\mathfrak{hs_3 nh2_{2+1}}$, the CS forms \eqref{hsCS2} and \eqref{hsCS2} provide actions involving the same fields \eqref{changeFieldsb}-\eqref{changeFieldsb2} as the Bargmann case. One can implement the rescaling \eqref{rescNRfields}, which now looks like
\begin{equation}\label{changeFieldsn}
\begin{aligned}
& \tau\rightarrow\ell^{-1}\, \tau,&\quad& e^a\rightarrow\ell^{-1}\, e^a,&\quad& m\rightarrow\ell^{-1}\, m,
\\
&V^a\rightarrow\ell^{-1}\, V^a,&\quad& E^{ab}\rightarrow\ell^{-1}\, E^{ab}, &\quad& M^a\rightarrow\ell^{-1}\, M^a,
\\
&V^{ab}\rightarrow\ell^{-1}\, V^{ab},&\quad& E^a\rightarrow\ell^{-1}\, E^a,&\quad& M^{ab}\rightarrow\ell^{-1}\, M^{ab},
\end{aligned}
\end{equation}
to find the CS forms given in Eqs. \eqref{hsCSNH1} and \eqref{hsCSNH2} of Appendix \ref{appNHB}.

\item \textbf{Spin-$3$ AdS-Lorentz algebras}: evaluating the action \eqref{CSNWexp} for $S^{(2)}_M$ yields the NR AdS-Lorentz CS gravity theory
first considered in Refs. \cite{Concha:2019lhn,Penafiel:2019czp}. The spin-$3$ extensions defined by and \eqref{hsCS2} and \eqref{hsCS2} lead to CS theories invariant under $\mathfrak{hs_3 nr\textnormal{-}adslor1_{2+1}}$ and $\mathfrak{hs_3 nr\textnormal{-}adslor2_{2+1}}$, respectively. These involve the gauge fields \eqref{changeFieldsb}-\eqref{changeFieldsm2}, \eqref{changeFieldsm}-\eqref{changeFieldsm2} appearing in the Maxwellian case, and reduces to it when applying the flat limit $\ell\rightarrow\infty$, provided the fields are rescaled as in Eq. \eqref{rescNRfields}, i.e.
\begin{equation}\label{changeFieldsal}
\begin{aligned}
&k^a\rightarrow\ell^{-2}\, k^a,&\quad& y\rightarrow\ell^{-2}\, y,&\quad& t\rightarrow\ell^{-2}\, t,
\\
&K^{ab}\rightarrow\ell^{-2}\, K^{ab},&\quad& Y^a\rightarrow\ell^{-2}\, Y^a,&\quad& T^a\rightarrow\ell^{-2}\, T^a,
\\
&K^a\rightarrow\ell^{-2}\, K^a,&\quad& Y^{ab}\rightarrow\ell^{-2}\, Y^{ab},&\quad& T^{ab}\rightarrow\ell^{-2}\, T^{ab}.
\end{aligned}
\end{equation}
The explicit form of the CS forms is given in Eqs. \eqref{hsCSNRAdSL1} and \eqref{hsCSNRAdSL2} of Appendix \ref{appAdSLM}.
\end{itemize}

\noindent Naturally, all the CS forms here considered can be obtained from the corresponding relativistic CS forms constructed out of the expanded/extended $\mathfrak{sl}(3,\mathbb R)$ algebras appearing in \eqref{diagexpsltw}, where the rescaling of the relativistic gauge fields in terms of $c$ can be easily read off from \eqref{redefsl3iexp} and \eqref{redefsl3iiexp} by demanding that the relativistic gauge connection matches the NR ones given in Eq. \eqref{expAs} .

\section{Concluding Remarks}
We have constructed NR higher-spin theories in $2+1$ space-time dimensions as expansions of particular higher-spin extensions of the Nappi-Witten algebras, namely, $\mathfrak{hs_3 nw1}$ and $\mathfrak{hs_3 nw2}$. These extended Nappi-Witten symmetries were obtained after applying In\"on\"u-Wigner contractions to suitable extensions of $\mathfrak{sl}(3,\mathbb R)$:
\begin{equation}\label{discussion1}
\begin{aligned}
&\mathfrak{sl}(3,\mathbb R)\oplus \mathfrak{so}(3)\;&
\underset{c\rightarrow\infty}{\longrightarrow}& \qquad\mathfrak{hs_3 nw1}
\\[6pt]
&\mathfrak{sl}(3,\mathbb R)\oplus \mathfrak{sl}(2,\mathbb R)\oplus \mathfrak{so}(2)\;&\underset{c\rightarrow\infty}{\longrightarrow}& \qquad\mathfrak{hs_3 nw2}
\end{aligned}
\end{equation}
The expansion method applied to these algebras gave rise to two infinite families of NR spin-$3$ symmeties, which are labeled by the coefficient $n$ entering in the semigroups $S^{(n)}_E$ and $S^{(n)}_M$ used in the expansion mechanism.
\begin{align}
&\textnormal{Generalized spin-}3\;\textnormal{Bargmann algebras}:& &
S_E^{(n)}\times \mathfrak{hs_3 nw1,2}, \label{genbarg}\\[9pt]
&\textnormal{Generalized spin-}3\;\textnormal{Newton-Hooke algebras}:&&
S_M^{(n)}\times \mathfrak{hs_3 nw1,2}.\; \label{gennh}
\end{align}
The first level of these families of algebras reproduces the higher spin extensions of the Extended Bargmann and Newton-Hooke algebras in three space-time dimensions previously found in \cite{Bergshoeff:2016soe}, whereas the $n=2$ case yields novel spin-$3$ NR symmetries in $2+1$-dimensions, which come to extend the (exotic) NR Maxwell algebra \cite{Aviles:2018jzw} and the NR AdS algebra \cite{Penafiel:2019czp}. Greater values of $n$ correspond to spin-$3$ extensions of Generalized Galilean symmetries algebras \cite{Gonzalez:2016xwo,Penafiel:2019czp}. We have shown that these two families of NR spin-$3$ algebras are related by a vanishing cosmological constant limit, i.e.
\begin{equation}
\begin{aligned}
&S^{(1)}_M\times\mathfrak{hs_3 nw1,2}\simeq\mathfrak{hs_3 nh1,2_{2+1}} &\xrightarrow[\ell\rightarrow\infty]\, &\qquad S^{(1)}_E\times\mathfrak{hs_3 nw1,2}\simeq\mathfrak{hs_3 barg1,2_{2+1}}\\
&S^{(2)}_M\times\mathfrak{hs_3 nw1,2}\simeq\mathfrak{hs_3 nr\textnormal{-}adslor1,2_{2+1}}&\xrightarrow[\ell\rightarrow\infty]\, &\qquad S^{(2)}_E\times\mathfrak{hs_3 nw1,2}\simeq\mathfrak{hs_3 nr\textnormal{-}max1,2_{2+1}}\\
&&\vdots&\\
&S^{(n)}_M\times\mathfrak{hs_3 nw1,2}&\xrightarrow[\ell\rightarrow\infty]\, &\qquad S^{(n)}_E\times\mathfrak{hs_3 nw1,2}\\
\end{aligned}
\end{equation}
Moreover, we have shown that all the algebras of this form are the NR version of an expanded $\mathfrak{sl}(3,\mathbb R)$ symmetry \cite{Caroca:2017izc}. Though, the relativistic algebras needed to define a $c\rightarrow\infty$ contraction leading to a non-degenerate invariant tensor, are semidirect sum extensions of such expanded $\mathfrak{sl}(3,\mathbb R)$ algebras. Unlike the known contractions giving rise to double and triple central extension in the spin-$2$ case (see for example \cite{Bergshoeff:2016lwr,Aviles:2018jzw,Concha:2019lhn}), these extensions do not simply add Abelian generators to the relativistic symmetries, but include Euclidean and Lorentzian symmetries in three dimensions (see \eqref{fulldiagram1ads}, \eqref{fulldiagram2ads} and \eqref{fulldiagram1adslor}, \eqref{fulldiagram2adslor}). In this way, the expansion mechanism produces a non-degenerate invariant bilinear form for every NR expanded algebra, together with a CS higher-spin gravity action. The particular cases $n=1,2$ were explicitly worked out.
\\\\
\noindent An interesting question for future research is the generalization of our results to the case of NR spin-$N$ symmetries for $N>3$. Indeed the fact that CS forms based on the $\mathfrak{sl}(N)\times \mathfrak{sl}(N)$ algebra define more general higher-spin gravity theories in $2+1$ dimensions \cite{Campoleoni:2010zq} admitting a flat limit \cite{Campoleoni:2016vsh}, together with the structure of the algebras $\mathfrak{sl}(3,\mathbb R)\oplus \mathfrak{so}(3)$ and $\mathfrak{sl}(3,\mathbb R)\oplus \mathfrak{sl}(2,\mathbb R)\oplus \mathfrak{so}(2)$
suggest that spin-$N$ extensions of the Nappi-Witten algebra $\mathfrak{nw}$ could be obtained as contractions of suitable extensions of $\mathfrak{sl}(N)$. A direct generalization of the structure \eqref{discussion1} signals the symmetries
\begin{equation}\label{discussion2}
\mathfrak{sl}(N,\mathbb R)\oplus \mathfrak{so}(N)
,\qquad
\mathfrak{sl}(N,\mathbb R)
\oplus
\mathfrak{sl}(N-1,\mathbb R)
\oplus\cdots\oplus
\mathfrak{sl}(3,\mathbb R)\oplus
\mathfrak{sl}(2,\mathbb R)\oplus \mathfrak{so}(2),
\end{equation}
as potential candidates to be expanded. One could as well explore extensions of $\mathfrak{sl}(N,\mathbb R)$ that look like
\begin{equation}
\begin{aligned}
\mathfrak{sl}(N,\mathbb R)
\oplus
\mathfrak{sl}(N-1,\mathbb R)
\oplus\cdots\oplus
\mathfrak{sl}(N-\mathfrak{m},\mathbb R)\oplus \mathfrak{so}(N-\mathfrak{m})
\;&\underset{c\rightarrow\infty}{\longrightarrow}& \quad\mathfrak{hs}_N \mathfrak{nw}(\mathfrak{1+m}),
\end{aligned}
\end{equation}
and thus interpolate between the ones given in \eqref{discussion2}, which could lead to several possible higher-spin extensions of the Nappi-Witten algebra. We hope to explore this problem elsewhere.\\

\noindent Due to the recently discovered relations between NR geometry and condensed matter systems \cite{Duval:2001hu,Son:2008ye,Geracie:2014nka,Gromov:2014vla}, and in particular, the recent application of CS theories invariant under extended Nappi-Witten algebras to construct effective descriptions of the fractional quantum Hall effect \cite{Salgado-Rebolledo:2021wtf,Salgado-Rebolledo:2021nfj}, it would be interesting to consider the NR spin-$3$ CS theories here obtained in the description of collective excitations in quantum Hall systems along the lines of \cite{Cappelli:2015ocj}, where NR higher-spin fields in three space-time dimensions appear in the multipole expansion of the low-energy excitations of a quantum Hall fluid. Moreover, in \cite{Salgado-Rebolledo:2021nfj}, a supersymmetric extension of the Nappi-Witten algebra was considered in the description of the Moore-Read fractional quantum Hall state. Thus, supersymmetric extensions of the $\mathfrak{hs_3 nw2}$ and $\mathfrak{hs_3 nw2}$ algebras defined along the same lines can be of interest in the exploration of collective excitations of the fractional quantum Hall effect in terms of superpartners \cite{Gromov:2019cgu}. Along the same lines, a natural generalization of our results is the inclusion of a space-time boundary and the construction of (Hamiltonian reductions of) higher-spin WZW theories (along the lines of, for example, \cite{Coussaert:1995zp}), which could lead to interesting edge dynamics and could be usfeul both in the context of the holographic duality in 3D gravity as well as as in condensed matter physics. We expect to consider this possibility in a future work.

\section*{Acknowledgements}
We are grateful to Giandomenico Palumbo for enlightening discussions. P.S-R thanks Joaquim Gomis for pointing out Ref. \cite{Boulanger:2002bt}, where Lie algebra expansions based on associative commutative algebras were considered. We specially thank Michelle Lacoste Adunka for carefully reading the manuscript. R.C. is partially funded by the National Agency for Research and Development ANID (ex-CONICYT) - FONDECYT grant N$^\circ$1211077 and by the Research Project Code DIREG$\_$09/2020 of the Universidad Cat\'olica de la Sant\'isima Concepci\'on, Chile. R.C. would like to thank the Direcci\'on de Investigaci\'on and the Vicerrector\'ia de Investigaci\'on of the Universidad Cat\'olica de la Sant\'isima Concepci\'on, Chile, for their constant support. P.S-R. has received funding from the Norwegian Financial Mechanism 2014-2021 via the National Science Centre (NCN) POLS grant 2020/37/K/ST3/03390.

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