text:
\section{Introduction}

$W$-algebras form an important class of vertex algebras.
They have been widely studied with connections to various areas of mathematics and mathematical physics such as two-dimensional conformal field theories, integrable systems, and four-dimensional gauge theories.
We refer to Arakawa's ICM talk \cite{MR3966808} and references therein for recent developments.
The goal of this paper is to relate $W$-algebras for rectangular nilpotent elements in type A to certain quantum groups, called affine Yangians.

The $W$-algebra $\mathcal{W}^{\kappa}(\mathfrak{g},f)$ is defined by the quantized Drinfeld-Sokolov reduction \cite{MR1071340, MR2013802}
from the following data:
a reductive Lie algebra $\mathfrak{g}$ over $\mathbb{C}$,
an invariant symmetric bilinear form $\kappa$ on $\mathfrak{g}$,
and a nilpotent element $f$ in $\mathfrak{g}$.
Here we omit to mention a choice of a grading of $\mathfrak{g}$.
It admits the Miura map, an injective homomorphism from $\mathcal{W}^{\kappa}(\mathfrak{g},f)$ to the universal affine vertex algebra $V^{\tilde\kappa}(\mathfrak{g}_{0})$ where $\mathfrak{g}_{0}$ denotes a Levi subalgebra of $\mathfrak{g}$ determined by a grading of $\mathfrak{g}$, and $\tilde{\kappa}$ is a bilinear form on $\mathfrak{g}_{0}$ induced from $\kappa$ with an appropriate shift.
Thus we can regard $\mathcal{W}^{\kappa}(\mathfrak{g},f)$ as a vertex subalgebra of  $V^{\tilde\kappa}(\mathfrak{g}_{0})$.

In this paper, we consider the case $\mathfrak{g} = \mathfrak{gl}_{N}$ for $N=ln$ with $l \geq 1$ and $n \geq 3$, and $f$ is a nilpotent element whose Jordan form corresponds to the partition $(l^{n})$.
We call them rectangular $W$-algebras because of the shape of the Young diagram corresponding to the partition $(l^{n})$.
Recently the rectangular $W$-algebras have been intensively studied; see \cite{MR3925243, MR4032024, MR4061286, MR4088277} for instance.

For a suitable choice of $\kappa$ (see Section~\ref{section:Rectangular_W-algebras}), the universal affine vertex algebra $V^{\tilde\kappa}(\mathfrak{g}_{0})$ is isomorphic to the tensor product of $l$ copies of the universal affine vertex algebra of $\mathfrak{gl}_{n}$.
Arakawa-Molev \cite{MR3598875} constructed elements $W_{i,j}^{(r)}$ ($r=1,\ldots,l$ and $i,j=1,\dots,n$) in $V^{\tilde\kappa}(\mathfrak{g}_{0}) \cong V^{\tilde\kappa}(\mathfrak{gl}_{n})^{\otimes l}$ and proved that they generate the rectangular $W$-algebra $\mathcal{W}^{\kappa}(\mathfrak{g},f)$.

Similar, and more refined story has been known for the corresponding finite $W$-algebras.
The rectangular finite $W$-algebra for $\mathfrak{g} = \mathfrak{gl}_{N}$ and $f = (l^{n})$ is regarded as a subalgebra of $U(\mathfrak{g}_{0}) \cong U(\mathfrak{gl}_{n})^{\otimes l}$.
Ragoucy-Sorba \cite{MR1700166} first noticed a relation between the rectangular finite $W$-algebras and Yangians of finite type both in type A.
Later Brundan-Kleshchev \cite{MR2199632} extended Ragoucy-Sorba's result; they constructed surjective algebra homomorphisms from shifted Yangians to finite $W$-algebras for arbitrary nilpotent elements in $\mathfrak{gl}_{N}$ and determined their kernels.
In \cite[Section~12]{MR2199632}, it was observed that in the case of rectangular type, the homomorphism is related to the coproduct and the evaluation map for the Yangian $Y(\mathfrak{gl}_{n})$.

Our main theorem is an affine analog of this observation, which asserts that the composition of the coproduct and the evaluation map for the affine Yangian $Y(\hat{\mathfrak{sl}}_n)$ gives an algebra homomorphism to the current algebra of the rectangular $W$-algebra $\mathcal{W}^{\kappa}(\mathfrak{g},f)$.
It answers affirmatively to an question raised by Genra in \cite[Introduction]{MR4091897}.

Let us state the main result and its consequence.
Let $k$ be the level of the rectangular $W$-algebra $\mathcal{W}^{\kappa}(\mathfrak{g},f) = \mathcal{W}^{\kappa}(\mathfrak{gl}_{N}, (l^n))$ where $N=ln$ with $l \geq 1$ and $n \geq 3$.
We associate with a vertex algebra $V$, its current algebra $\mathfrak{U}(V)$.
Let $\hbar, \varepsilon$ be the parameters of the affine Yangian $Y(\hat{\mathfrak{sl}}_n)$.
We assume that the condition $k+N=-\varepsilon/\hbar$ is satisfied.
Then we will define a map $\Phi_{l}$ in Definition~\ref{dfn:map}, which is an algebra homomorphism from a completed affine Yangian $Y(\hat{\mathfrak{sl}}_n)_{\mathrm{comp}}$ to a completed tensor product $U(\hat{\mathfrak{gl}}_{n}^{\alpha})_{{\mathrm{comp}}}^{\otimes l}$ of the universal enveloping algebras of the affine Lie algebras $\hat{\mathfrak{gl}}_{n}^{\alpha}$ at level $\alpha = k+N-n$.
Let $\Delta$ denote the coproduct of $Y(\hat{\mathfrak{sl}}_n)$.
Following \cite[Section~7.2]{MR4091897}, we will introduce in Section~\ref{section:Coproduct_and_parabolic_induction} an injective algebra homomorphism
\[
	\tilde{\Delta}_{l_1,l_2} \colon \mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f)) \to \mathfrak{U}(\mathcal{W}^{\kappa_1}(\mathfrak{gl}_{N_1}, f_1)) \otimes_{\mathrm{comp}} \mathfrak{U}(\mathcal{W}^{\kappa_2}(\mathfrak{gl}_{N_2}, f_2))
\]
with $l=l_1+l_2$ and $N_i=l n_i$, $f_i=(l_i^n)$.

\begin{thm}[Theorem~\ref{thm:main}, Corollary~\ref{cor:parabolic_induction}]\label{thm:introduction_main}
Assume $k+N=-\varepsilon/\hbar$.
Then the map $\Phi_{l}$ gives an algebra homomorphism
\[
	\Phi_{l} \colon Y(\hat{\mathfrak{sl}}_n)_{\mathrm{comp}} \to \mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f))
\]
which makes the following diagram commutative:
\[
	\xymatrix{
		Y(\hat{\mathfrak{sl}}_n)_{\mathrm{comp}} \ar[d]_{\Delta} \ar[r]^{\Phi_{l}} \ & \ \mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f)) \ar[d]^{\tilde{\Delta}_{l_1,l_2}} \\
		Y(\hat{\mathfrak{sl}}_n)^{\otimes 2}_{\mathrm{comp}} \ar[r]^-{\Phi_{l_1} \otimes \Phi_{l_2}} \ \ & \ \ \mathfrak{U}(\mathcal{W}^{\kappa_1}(\mathfrak{gl}_{N_1}, f_1)) \otimes_{\mathrm{comp}} \mathfrak{U}(\mathcal{W}^{\kappa_2}(\mathfrak{gl}_{N_2}, f_2)).
	}
\]
Moreover, $\Phi_{l}$ is surjective if $k+N-n \neq 0$.
\end{thm}

Given representations $M_i$ of $\mathcal{W}^{\kappa_i}(\mathfrak{gl}_{N_i}, f_i)$ for $i=1,2$, we regard $M_1 \otimes M_2$ as a representation of $\mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f))$ via $\tilde{\Delta}_{l_1,l_2}$.
This is called parabolic induction.
By Theorem~\ref{thm:introduction_main}, each $M_1$, $M_2$, and $M_1 \otimes M_2$ are regarded as representations of the affine Yangian $Y(\hat{\mathfrak{sl}}_n)$ via $\Phi_{l_1}$, $\Phi_{l_2}$, and $\Phi_{l}$ respectively.
The commutative diagram in Theorem~\ref{thm:introduction_main} means that $M_1 \otimes M_2$ is the tensor product representation via the coproduct $\Delta$.

We obtain a similar result for the super setting in Theorem~\ref{thm:introduction_main_super}.
The second named author constructed in \cite{Affine_super_Yangians_and_rectangular_W-superalgebras} an algebra homomorphism $\Phi$ from $Y(\hat{\mathfrak{sl}}_n)_{\mathrm{comp}}$ to $\mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f))$ directly.
Namely, he assigned elements of $\mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f))$ to generators of $Y(\hat{\mathfrak{sl}}_n)$ and checked that it respects the defining relations of $Y(\hat{\mathfrak{sl}}_n)$.
His construction is applicable also to the super setting.
The map $\Phi_{l}$ coincides with $\Phi$ up to a twist in both the non-super and the super case (see Remark~\ref{rem:difference} (i) for the twist in the non-super case).
Hence Theorem~\ref{thm:introduction_main} and Theorem~\ref{thm:introduction_main_super} give an alternative proof of the main result of \cite{Affine_super_Yangians_and_rectangular_W-superalgebras}.

Our work is motivated by the Alday-Gaiotto-Tachikawa (AGT) correspondence for parabolic sheaves on $\mathbb{P}^{1} \times \mathbb{P}^{1}$.
There are various versions of AGT correspondence and we only refer to the original paper \cite{MR2586871} and \cite[Introduction, 1.5 and 1.11 (4)]{MR3592485} here.
It has been shown that the affine Yangian $Y(\hat{\mathfrak{sl}}_n)$ acts on the localized equivariant cohomology of affine Laumon spaces (framed moduli spaces of parabolic sheaves) by Feigin-Finkelberg-Negut-Rybnikov \cite{MR2827177} and Finkelberg-Tsymbaliuk \cite{MR3971731}.
We conjecture that our map $\Phi_{l}$ induces an action of $\mathcal{W}^{\kappa}(\mathfrak{g},f)$ on the equivariant cohomology of affine Laumon spaces and that the action is one predicted by the AGT correspondence.
This conjecture would give an affine analog of results of Braverman-Feigin-Finkelberg-Rybnikov \cite{MR2851149} and Nakajima \cite{MR3024827} in the case of rectangular type.
We note that Negut studies the $K$-theory version in \cite{Toward_AGT_for_parabolic_sheaves, Deformed_W_algebras_in_type_A_for_rectangular_nilpotent} by a different approach.

We finish this section with comments on the case for $n=1,2$.
Works of Maulik-Okounkov \cite{MR3951025} and Schiffmann-Vasserot \cite{MR3150250} related the affine Yangian $Y(\hat{\mathfrak{gl}}_1)$ to the principal $W$-algebras of type A, and established the AGT correspondence in this case.
We are missing the $n=2$ case at the moment since neither of coproduct nor evaluation map for $Y(\hat{\mathfrak{sl}}_2)$ has not been defined.

This paper is organized as follows.
In Section~\ref{section:Rectangular_W-algebras}--\ref{section:Parabolic_induction}, we prepare basics on the rectangular $W$-algebras.
In Section~\ref{section:Affine_Yangians}--\ref{section:Evaluation_map}, we recall the coproduct and the evaluation map for the affine Yangians.
We construct the algebra homomorphisms from the affine Yangians to the rectangular $W$-algebras in Section~\ref{section:Affine_Yangians_and_rectangular_W-algebras} and relate the parabolic induction for the rectangular $W$-algebras to the coproduct for the affine Yangians in Section~\ref{section:Coproduct_and_parabolic_induction}.
We discuss some commuting elements of the rectangular $W$-algebras in Section~\ref{section:Commuting_elements}.
In Section~\ref{section:Super_case}, we give a brief explanation for the super case.
Appendix A, B, and C collect some computations.

\subsection*{Acknowledgments}
The authors are grateful to Tomoyuki Arakawa, Boris Feigin, Ryo Fujita, Naoki Genra,
Toshiro Kuwabara, Andrew Linshaw, Hiraku Nakajima, Shigenori Nakatsuka, Andrei Negut, Masatoshi Noumi, Shoma Sugimoto, Husileng Xiao, Yasuhiko Yamada, and Shintarou Yanagida for valuable discussions and suggestions.

Some part of results of this paper were presented by the first named author in Workshop on 3d Mirror Symmetry and AGT Conjecture held in October 21--25, 2019 at Institute for Advanced Study in Mathematics, Zhejiang University, Hangzhou.
He thanks to their hospitality.

The first named author was supported by JSPS KAKENHI Grant Number 18K13390, 21K03155.
His work was also supported in part by JSPS Bilateral Joint Projects (JSPS-RFBR collaboration) ``Elliptic algebras, vertex operators and link invariants'' from MEXT, Japan.
The second named author was supported by Grant-in-Aid for JSPS Fellows 20J12072.

\section*{Notations for parameters}

Throughout the paper, we fix nonzero positive integers $l$ and $n$, then put $N=ln$.
We assume $n \geq 3$ from Section~\ref{section:Affine_Yangians} to Section~\ref{section:Commuting_elements}, in particular in the main theorem.

We fix a complex number $k$, which stands for the level of the $W$-algebra.
We also use an alternative parameter $\alpha$ defined by $\alpha= k+(l-1)n= k+N-n$ except for Section~\ref{section:Super_case}.

\section{Rectangular $W$-algebras}\label{section:Rectangular_W-algebras}

The $W$-algebra $\mathcal{W}^{\kappa}(\mathfrak{g},f)$ is defined by the quantized Drinfeld-Sokolov reduction from the data consisting of a reductive Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ equipped with a $\mathfrak{g}$-invariant symmetric bilinear form $\kappa$
and a nilpotent element $f$ in $\mathfrak{g}$.
In this paper, we do not use its definition but a characterization by the Miura map, which will be recalled in Section~\ref{section:Miura_map}.
See \cite{MR2013802} for the definition.

Let $\mathfrak{g} = \mathfrak{gl}_{N}$ be the complex general linear Lie algebra consisting of $N \times N$ matrices.
We denote by $E_{i,j}$ the matrix unit whose $(i,j)$-entry is $1$.
Put $I_{N} = \sum_{i=1}^{N} E_{i,i}$ and $\mathfrak{z}_{N} = \mathbb{C} I_{N}$.
Then we have a decomposition
\[
	\mathfrak{g} = \mathfrak{sl}_{N} \oplus \mathfrak{z}_{N}.
\]
Fix $k \in \mathbb{C}$.
We define an invariant symmetric bilinear form $\kappa$ on $\mathfrak{g}$ by
\begin{equation}
	\kappa(X,Y) = \begin{cases}
		k \tr(XY) & \text{if } X,Y \in \mathfrak{sl}_{N},\\
		(k+N) \tr(XY) & \text{if } X,Y \in \mathfrak{z}_{N},\\
		0 & \text{if } X \in \mathfrak{sl}_{N} \text{ and } Y \in \mathfrak{z}_{N}.
	\end{cases} \label{eq:kappa}
\end{equation}
The explicit value of $\kappa$ is
\[
	\kappa(E_{i,j},E_{p,q}) = \delta_{iq}\delta_{jp}k + \delta_{ij}\delta_{pq}.
\]

We assume $N= l n$ with nonzero positive integers $l$ and $n$.
As vector spaces, we have an isomorphism
\begin{equation}
	\mathfrak{gl}_{l} \otimes \mathfrak{gl}_{n} \cong \mathfrak{g}=\mathfrak{gl}_{N}, \qquad E_{s,t} \otimes E_{i,j} \mapsto E_{(s-1)n+i,(t-1)n+j}. \label{eq:isom_gl}
\end{equation}
We define a $\mathbb{Z}$-grading of $\mathfrak{gl}_{l}$ by $\deg E_{s,t} = t-s$ and denote by $(\mathfrak{gl}_{l})_{j}$ its degree $j$ component.
It induces a $\mathbb{Z}$-grading of $\mathfrak{g}$ by
\[
	\mathfrak{g}_{j} = (\mathfrak{gl}_{l})_{j} \otimes \mathfrak{gl}_{n}
\]
using the identification (\ref{eq:isom_gl}).
In particular, we have $\mathfrak{g}_{0} \cong \mathfrak{gl}_{n}^{\oplus l}$.
We set
\begin{equation}
	E_{i,j}^{[s]} = E_{(s-1)n+i,(s-1)n+j} \label{eq:diagonal}
\end{equation}
for $s=1,\ldots,l$ and $i,j=1,\ldots,n$, then $\{ E_{i,j}^{[s]} \mid i,j=1,\ldots,n \}$ forms a $\mathbb{C}$-basis of the $s$-th component of $\mathfrak{g}_{0} \cong \mathfrak{gl}_{n}^{\oplus l}$.
Set
\begin{equation}
	\begin{split}
		f = \left( \sum_{s=1}^{l-1} E_{s+1,s} \right) \otimes I_{n} = \sum_{s=1}^{l-1} \sum_{i=1}^{n} E_{sn+i,(s-1)n+i} = \left[ \begin{array}{ccccc}
			O &  & & & O\\
			I_{n} & O & & & \\
			 & I_{n} & \dots & & \\
			 &  & & O & \\
			O &  & &  I_{n} & O
		\end{array} \right]. \label{eq:nilpotent}
	\end{split}
\end{equation}
The Jordan form of the nilpotent element $f$ corresponds to the partition $(l^{n})$ of rectangular type.

From the data ($\mathfrak{g}, \kappa, f$) above, we can define the rectangular $W$-algebra $\mathcal{W}^{\kappa}(\mathfrak{g},f)$ as a vertex algebra.
It is isomorphic to the tensor product
\[
	\mathcal{W}^{k}(\mathfrak{sl}_{N},f) \otimes V^{k+N}(\mathfrak{z}_{N}),
\]
where $\mathcal{W}^{k}(\mathfrak{sl}_{N},f)$ is the $W$-algebra associated with $\mathfrak{sl}_{N}$ and $f$ at level $k$,
and $V^{k+N}(\mathfrak{z}_{N})$ is the Heisenberg vertex algebra associated with $\mathfrak{z}_{N}$ at level $k+N$.

\section{Affine Lie algebras}\label{section:Affine_Lie_algebras}

Let $\kappa_{\mathfrak{g}}^{\circ}$ and $\kappa_{\mathfrak{g}_{0}}^{\circ}$ be the Killing forms on $\mathfrak{g}$ and $\mathfrak{g}_{0}$ respectively.
We define an invariant symmetric bilinear form $\tilde{\kappa}$ on $\mathfrak{g}_{0}$ by
\[
	\tilde{\kappa} = \kappa |_{\mathfrak{g}_{0}} + \dfrac{1}{2} ( \kappa_{\mathfrak{g}}^{\circ} |_{\mathfrak{g}_{0}} - \kappa_{\mathfrak{g}_{0}}^{\circ} ).
\]
This bilinear form is often denoted by $\tau_{\kappa}$, but we do not use the notation in this paper.
The explicit value of $\tilde{\kappa}$ is
\begin{equation}
	\tilde{\kappa}(E_{i,j}^{[s]},E_{p,q}^{[t]}) = \delta_{st} \Big( \delta_{iq}\delta_{jp}(k+(l-1)n) + \delta_{ij}\delta_{pq} \Big). \label{eq:tildekappa}
\end{equation}

For an arbitrary $\alpha \in \mathbb{C}$, we can define an invariant symmetric bilinear form $(\, , \,)$ on $\mathfrak{gl}_{n}$ by replacing $k$ and $N$ with $\alpha$ and $n$ in the definition of $\kappa$ in (\ref{eq:kappa});
\[
	(X,Y) = \begin{cases}
		\alpha \tr(XY) & \text{if } X,Y \in \mathfrak{sl}_{n},\\
		(\alpha+n) \tr(XY) & \text{if } X,Y \in \mathfrak{z}_{n},\\
		0 & \text{if } X \in \mathfrak{sl}_{n} \text{ and } Y \in \mathfrak{z}_{n}.
	\end{cases}
\]
Its explicit value is
\begin{equation}
	(E_{i,j},E_{p,q}) = \delta_{iq}\delta_{jp}\alpha + \delta_{ij}\delta_{pq}. \label{eq:gln_bilinear_form}
\end{equation}
Hence, if we put
\[
	\alpha = k+(l-1)n
\]
and compare (\ref{eq:tildekappa}) and (\ref{eq:gln_bilinear_form}), then the restriction of $\tilde{\kappa}$ to each component of $\mathfrak{g}_{0} \cong \mathfrak{gl}_{n}^{\oplus l}$ is identified with $(\, , \,)$.
We use the same symbol $\tilde{\kappa}$ for this bilinear form on $\mathfrak{gl}_{n}$ by abuse of notation.

We define the affine Lie algebra $\hat{\mathfrak{g}}_{0}$ associated with $\tilde{\kappa}$ by
\[
	\hat{\mathfrak{g}}_{0} = \mathfrak{g}_{0}[t,t^{-1}] \oplus \mathbb{C} \mathbf{1},
\]
\[
	[X t^{m}, Y t^{m'}] = [X,Y] t^{m+m'} + m \delta_{m+m',0} \tilde{\kappa}(X,Y) \mathbf{1}, \quad \text{$\mathbf{1}$ is central}.
\]
Similarly we define the affine Lie algebra $\hat{\mathfrak{gl}}_{n}$ associated with $\tilde{\kappa}$ by
\[
	\hat{\mathfrak{gl}}_{n} = \mathfrak{gl}_{n}[t,t^{-1}] \oplus \mathbb{C} \mathbf{1}',
\]
\[
	[X t^{m}, Y t^{m'}] = [X,Y] t^{m+m'} + m \delta_{m+m',0} \tilde{\kappa}(X,Y) \mathbf{1}', \quad \text{$\mathbf{1}'$ is central}.
\]
We have an isomorphism
\[
	\hat{\mathfrak{g}}_{0} / (\mathbf{1}-1) \cong \left( \hat{\mathfrak{gl}}_{n} / (\mathbf{1}'-1) \right)^{\oplus l}.
\]
We put
\[
	\hat{\mathfrak{gl}}_{n}^{\alpha} = \hat{\mathfrak{gl}}_{n} / (\mathbf{1}'-1).
\]
It is the direct sum of the affine Lie algebra $\hat{\mathfrak{sl}}_{n}$ at level $\alpha$ and the Heisenberg Lie algebra $\hat{\mathfrak{z}}_{n}$ at level $\alpha+n$.

We consider the universal affine vertex algebras
\[
	V^{\tilde{\kappa}}(\mathfrak{g}_{0}) = U(\hat{\mathfrak{g}}_{0}) \otimes_{U(\mathfrak{g}_{0}[t] \oplus \mathbb{C} \mathbf{1})} \mathbb{C}, \quad V^{\tilde{\kappa}}(\mathfrak{gl}_{n}) = U(\hat{\mathfrak{gl}}_{n}) \otimes_{U(\mathfrak{gl}_{n}[t] \oplus \mathbb{C} \mathbf{1}')} \mathbb{C},
\]
where $\mathbb{C}$ denotes both the one-dimensional representations defined by
\[
	\mathfrak{g}_{0}[t], \mathfrak{gl}_{n}[t] \mapsto 0, \quad \mathbf{1}, \mathbf{1}' \mapsto 1.
\]
We have an isomorphism
\[
	V^{\tilde{\kappa}}(\mathfrak{g}_{0}) \cong V^{\tilde{\kappa}}(\mathfrak{gl}_{n})^{\otimes l}
\]
of vertex algebras.
We also have an isomorphism
\[
	V^{\tilde{\kappa}}(\mathfrak{gl}_{n}) \cong V^{\alpha}(\mathfrak{sl}_{n}) \otimes V^{\alpha+n}(\mathfrak{z}_{n}),
\]
where $V^{\alpha}(\mathfrak{sl}_{n})$ is the universal affine vertex algebra associated with $\hat{\mathfrak{sl}}_{n}$ at level $\alpha$, and $V^{\alpha+n}(\mathfrak{z}_{n})$ is the Heisenberg vertex algebra associated with $\hat{\mathfrak{z}}_{n}$ at level $\alpha+n$.

We denote the element $X t^{m}$ by $X(m)$.
As vector spaces, we have
\[
	V^{\tilde{\kappa}}(\mathfrak{g}_{0}) \cong U(\mathfrak{g}_{0}[t^{-1}]t^{-1}) \ket{0},
\]
where $\ket{0} = 1 \otimes 1$ denotes the vacuum vector.
For $X \in \mathfrak{g}_{0}$ and $m \in \mathbb{Z}_{\geq 1}$, we use the symbol $X[-m]$ for the element $X(-m)\ket{0}$ of $V^{\tilde{\kappa}}(\mathfrak{g}_{0})$.

We associate with a vertex algebra $V$ the current algebra $\mathfrak{U}(V)$ (sometimes called the universal enveloping algebra).
Roughly speaking, it is a completed associative algebra topologically generated by elements $vt^{m}$ ($v \in V$ and $m \in \mathbb{Z}$) corresponding to the Fourier coefficients $v_{(m)}$ of the generating field $v(z)=\sum_{m \in \mathbb{Z}} v_{(m)}z^{-m-1}$.
See \cite[Chapter~4]{MR2082709} or \cite[Section~3.11]{MR2318558} for details (it is denoted by $\widetilde{U}(V)$ in \cite{MR2082709}).

Consider the universal affine vertex algebra $V^{\tilde{\kappa}}(\mathfrak{gl}_{n})$.
Its current algebra $\mathfrak{U}(V^{\tilde{\kappa}}(\mathfrak{gl}_{n}))$ is identified with a completion $U(\hat{\mathfrak{gl}}_{n}^{\alpha})_{\mathrm{comp}}$ of $U(\hat{\mathfrak{gl}}_{n}^{\alpha})$ given as follows.
In general, let $A = \bigoplus_{d \in \mathbb{Z}} A_d$ be a $\mathbb{Z}$-graded algebra.
We call
\[
	A_{\mathrm{comp}} = \bigoplus_{d \in \mathbb{Z}} \lim_{\substack{\leftarrow\\ m}} \left( A_d / \sum_{r > m} A_{d-r} A_{r} \right)
\]
the standard degree-wise completion of $A$ according to \cite[A.2]{MR2318558}.
We define a grading of $U(\hat{\mathfrak{gl}}_{n}^{\alpha})$ by $\deg X(m) = m$.
Then we denote by $U(\hat{\mathfrak{gl}}_{n}^{\alpha})_{\mathrm{comp}}$ the standard degree-wise completion of $U(\hat{\mathfrak{gl}}_{n}^{\alpha})$.
Moreover $U(\hat{\mathfrak{gl}}_{n}^{\alpha})_{\mathrm{comp}}^{\otimes l}$ denotes the degree-wise completed tensor product of $U(\hat{\mathfrak{gl}}_{n}^{\alpha})$ defined in the same way.
Then we have
\begin{equation}
	\mathfrak{U}(V^{\tilde{\kappa}}(\mathfrak{g}_{0})) \cong U(\hat{\mathfrak{gl}}_{n}^{\alpha})_{\mathrm{comp}}^{\otimes l}. \label{eq:tensor}
\end{equation}
Let the symbol $X^{[s]}$ for $s=1,\ldots,l$ denote the element $1^{\otimes (s-1)} \otimes X \otimes 1^{\otimes (l-s)}$.
This notation is compatible with one in (\ref{eq:diagonal}) under the identification (\ref{eq:tensor}).

\section{Miura map and Arakawa-Molev's generators}\label{section:Miura_map}

The Miura map is an injective homomorphism
\begin{equation}
	\mathcal{W}^{\kappa}(\mathfrak{g},f) \to V^{\tilde{\kappa}}(\mathfrak{g}_{0}) \cong V^{\tilde{\kappa}}(\mathfrak{gl}_{n})^{\otimes l} \label{eq:Miura}
\end{equation}
of vertex algebras.
See \cite[Section~3]{MR3598875} for details.
Consider the current algebras of (\ref{eq:Miura}), then we have an injective homomorphism
\begin{equation}
	\mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f)) \to U(\hat{\mathfrak{gl}}_{n}^{\alpha})_{\mathrm{comp}}^{\otimes l} \label{eq:Miura_algebra}
\end{equation}
of associative algebras.
We also call (\ref{eq:Miura_algebra}) Miura map.
In the sequel, we identify $\mathcal{W}^{\kappa}(\mathfrak{g},f)$ with its image by the Miura map and regard it as a subalgebra of $V^{\tilde{\kappa}}(\mathfrak{gl}_{n})^{\otimes l}$.
Also we regard $\mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f))$ as a subalgebra of $U(\hat{\mathfrak{gl}}_{n}^{\alpha})_{\mathrm{comp}}^{\otimes l}$.

We recall generators of $\mathcal{W}^{\kappa}(\mathfrak{g},f)$ introduced by Arakawa-Molev~\cite{MR3598875}.
Let $A^{[s]} = \left( E_{i,j}^{[s]}[-1]_{(-1)} \right)_{1 \leq i,j \leq n}$ be the $n \times n$ matrix whose $(i,j)$-entry is $E_{i,j}^{[s]}[-1]_{(-1)}$.
Let $\tau$ be a formal operator satisfying
\[
	[X[-m], \tau] = \alpha m X[-m-1]
\]
for $X \in \mathfrak{g}_{0}$.
We define a matrix $W^{(r)}$ by
\begin{equation}
	(\tau + A^{[1]}) (\tau + A^{[2]}) \cdots (\tau + A^{[l]}) = \tau^{l} + \sum_{r=1}^{l} \tau^{l-r} W^{(r)}. \label{eq:Miura_transformation}
\end{equation}
In (\ref{eq:Miura_transformation}), if we obtain an element of the form $X_1[-m_1]_{(-1)} \cdots X_a[-m_a]_{(-1)}$, then we regard the right-most $X_a[-m_a]_{(-1)}$ as $X_a[-m_a]$.
Then the $(i,j)$-entry, denoted by $W_{i,j}^{(r)}$, of $W^{(r)}$ gives an element of $V^{\tilde{\kappa}}(\mathfrak{g}_{0})$.

\begin{thm}[\cite{MR3598875}, Theorem~3.1 and Corollary~3.2]
The elements $W_{i,j}^{(r)}$ belong to $\mathcal{W}^{\kappa}(\mathfrak{g},f)$ for all $r=1, \ldots, l$ and $i,j=1, \ldots, n$.
They freely generate $\mathcal{W}^{\kappa}(\mathfrak{g},f)$ as a vertex algebra.
\end{thm}

\begin{exam}
We have
\begin{align*}
	W_{i,j}^{(1)} &= \sum_{s=1}^{l} E_{i,j}^{[s]}[-1],\\
	W_{i,j}^{(2)} &= \sum_{1 \leq s_1 < s_2 \leq l} \sum_{a=1}^{n} E_{i,a}^{[s_1]}[-1]_{(-1)} E_{a,j}^{[s_2]}[-1] + \alpha \sum_{s=1}^{l} (l-s) E_{i,j}^{[s]}[-2].
\end{align*}
\end{exam}

\begin{rem}\label{rem:difference}
The elements $W_{i,j}^{(r)}$ are slightly different from ones in the literature.
\begin{enumerate}
\item
Let ${}^{\mathrm{AM}}W_{i,j}^{(r)}$ denote the image by the Miura map of the generators given in \cite[Corollary~3.2]{MR3598875}.
They are defined by
\begin{equation*}
	(- \tau + A^{[1]}) (- \tau + A^{[2]}) \cdots (- \tau + A^{[l]}) = (- \tau)^{l} + \sum_{r=1}^{l} \left( {}^{\mathrm{AM}}W_{j,i}^{(r)} \right)_{1 \leq i,j \leq n} (- \tau)^{l-r}.
\end{equation*}
We have an algebra automorphism $\sigma$ on $V^{\tilde{\kappa}}(\mathfrak{g}_{0})$ induced from $X^{[s]} \mapsto X^{[l-s+1]}$ for $X \in \mathfrak{gl}_{n}$.
Under the identification
\[
	V^{\tilde{\kappa}}(\mathfrak{g}_{0}) \cong V^{\tilde{\kappa}}(\mathfrak{gl}_{n})^{\otimes l},
\]
$\sigma$ corresponds to the operation which reverses the order of the tensor product.
Then we have
\[
	W_{i,j}^{(r)} = \sigma \left( {}^{\mathrm{AM}}W_{j,i}^{(r)} \right).
\]

\item
In \cite{MR4061286}, generators ${{U_{(r)}}^{a}}_{b}$ are used.
They are defined by
\begin{equation*}
	(\tau + A^{[1]}) (\tau + A^{[2]}) \cdots (\tau + A^{[l]}) = \tau^{l} + \sum_{r=1}^{l} \left( {{U_{(r)}}^{a}}_{b} \right)_{1 \leq a,b \leq n} \tau^{l-r}.
\end{equation*}
The relation with our generators $W_{i,j}^{(r)}$ for $r=1,2$ is given as follows:
\[
	W_{i,j}^{(1)} = {{U_{(1)}}^{i}}_{j}, \quad W_{i,j}^{(2)} = {{U_{(2)}}^{i}}_{j} + (l-1)\alpha \partial {{U_{(1)}}^{i}}_{j}.
\]
Formulas for OPE appearing in this paper can be obtained from ones given in \cite[Section~3.1]{MR4061286}.
\end{enumerate}
\end{rem}

The current algebra $\mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f))$ is an associative algebra which has topological generators
\[
	\{ W_{i,j}^{(r)}t^{m} \mid 1 \leq r \leq l, \ \ 1 \leq i,j \leq n, \ \ m \in \mathbb{Z} \},
\]
where the element $W_{i,j}^{(r)}t^{m}$ corresponds to the Fourier coefficient $(W_{i,j}^{(r)})_{(m)}$ of the element $W_{i,j}^{(r)}$ of $\mathcal{W}^{\kappa}(\mathfrak{g},f)$.
We set
\[
	W_{i,j}^{(r)}(m) = W_{i,j}^{(r)}t^{m+r-1}.
\]

\begin{exam}\label{exam:generators}
We have
\begin{align*}
	W_{i,j}^{(1)}(m) &= \sum_{s=1}^{l} E_{i,j}^{[s]}(m),\\
	W_{i,j}^{(2)}(m) &= \sum_{r \in \mathbb{Z}} \sum_{1 \leq s_1 < s_2 \leq l} \sum_{a=1}^{n} E_{i,a}^{[s_1]}(m+r) E_{a,j}^{[s_2]}(-r) - (m+1)\alpha \sum_{s=1}^{l} (l-s) E_{i,j}^{[s]}(m).
\end{align*}
\end{exam}

The elements $W_{i,j}^{(1)}(m)$ satisfy the following commutation relation:
\[
	[W_{i,j}^{(1)}(m), W_{p,q}^{(1)}(m')] = \delta_{pj} W_{i,q}^{(1)}(m+m') - \delta_{iq} W_{p,j}^{(1)}(m+m') + m \delta_{m+m',0} l (\delta_{iq}\delta_{jp}\alpha + \delta_{ij}\delta_{pq}).
\]
That is, they generate the affine Lie algebra of $\mathfrak{gl}_{n}$ associated with the bilinear form $l \times \tilde{\kappa}$.

The set $\{ W_{i,j}^{(r)} \mid 1 \leq r \leq l, \ 1 \leq i,j \leq n \}$ forms a free generator of $\mathcal{W}^{\kappa}(\mathfrak{g},f)$.
One can reduce the generators to the elements for $r=1,2$ under the assumption $n \geq 2$ and $\alpha \neq 0$, although they do not freely generate $\mathcal{W}^{\kappa}(\mathfrak{g},f)$.

\begin{prop}[\cite{Affine_super_Yangians_and_rectangular_W-superalgebras}]\label{prop:generators}
Assume $n \geq 2$ and $\alpha \neq 0$.
Then the elements $W_{i,j}^{(r)}$ for $r=1,2$ and all $i,j$ generate $\mathcal{W}^{\kappa}(\mathfrak{g},f)$ as a vertex algebra.
In particular, $W_{i,j}^{(r)}(m)$ for $r=1,2$ and all $i,j,m$ topologically generate $\mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f))$ as an associative algebra.
\end{prop}

\section{Parabolic induction}\label{section:Parabolic_induction}

Let us recall the parabolic induction for the rectangular $W$-algebras (see \cite[Introduction,  Theorem~A and B]{MR4091897} for a general result).

Choose integers $l_1, l_2 \geq 1$ satisfying $l=l_1+l_2$.
Set $N_1 = l_1 n$ and $N_2 = l_2 n$ so that $N=N_1+N_2$.
We also take $k_1$ and $k_2$ so that
\begin{equation}
	k+N = k_1+N_1 = k_2+N_2 \label{eq:relations_among_levels}
\end{equation}
holds.
Since the parameter $\alpha$ is defined by $k+N = \alpha + n$, the equality (\ref{eq:relations_among_levels}) means that $\alpha$ is unchanged if we replace $(k,l,N)$ with $(k_i,l_i,N_i)$ ($i=1,2$).

For $(k_i,l_i,N_i)$ ($i=1,2$), we define an invariant symmetric bilinear form $\kappa_{i}$ on $\mathfrak{gl}_{N_i}$ in a way similar to (\ref{eq:kappa}), and a nilpotent element $f_{i}$ as in (\ref{eq:nilpotent}) corresponding to the partition ($l_i^{n}$).
Associated with the data, $\mathcal{W}^{\kappa_1}(\mathfrak{gl}_{N_1},f_1)$ and $\mathcal{W}^{\kappa_2}(\mathfrak{gl}_{N_2},f_2)$ are defined.
We have the Miura maps
\[
	\mathcal{W}^{\kappa_i}(\mathfrak{gl}_{N_i},f_i) \to V^{\tilde{\kappa}_i}(\mathfrak{gl}_{n})^{\otimes l_i}
\]
for $i=1,2$.
Since $\alpha$ is taken uniformly for $(k,l,N)$ and $(k_i,l_i,N_i)$, $\tilde{\kappa}_i$ on $\mathfrak{gl}_{n}$ is the same as $\tilde{\kappa}$.
Hence $\mathcal{W}^{\kappa_1}(\mathfrak{gl}_{N_1},f_1) \otimes \mathcal{W}^{\kappa_2}(\mathfrak{gl}_{N_2},f_2)$ can be regarded as a subalgebra of $V^{\tilde{\kappa}}(\mathfrak{gl}_{n})^{\otimes l}$ by the tensor product of the Miura maps.
Then it turns out that $\mathcal{W}^{\kappa_1}(\mathfrak{gl}_{N_1},f_1) \otimes \mathcal{W}^{\kappa_2}(\mathfrak{gl}_{N_2},f_2)$ contains $\mathcal{W}^{\kappa}(\mathfrak{g},f)$ inside $V^{\tilde{\kappa}}(\mathfrak{gl}_{n})^{\otimes l}$.
We denote the inclusion by $\Delta_{l_1,l_2}$:
\[
	\Delta_{l_1,l_2} \colon \mathcal{W}^{\kappa}(\mathfrak{g},f) \to \mathcal{W}^{\kappa_1}(\mathfrak{gl}_{N_1},f_1) \otimes \mathcal{W}^{\kappa_2}(\mathfrak{gl}_{N_2},f_2).
\]

The generators given in Section~\ref{section:Miura_map} are related by $\Delta_{l_1,l_2}$ as follows.
Divide the left-hand side $(\tau + A^{[1]}) \cdots (\tau + A^{[l]})$ of (\ref{eq:Miura_transformation}) into two blocks:
\[
	(\tau + A^{[1]}) \cdots (\tau + A^{[l_1]}) \ \text{ and } \ (\tau + A^{[l_1+1]}) \cdots (\tau + A^{[l]}).
\]
Then we can similarly define generators for $\mathcal{W}^{\kappa_1}(\mathfrak{gl}_{N_1},f_1)$ and $\mathcal{W}^{\kappa_2}(\mathfrak{gl}_{N_2},f_2)$, and we can relate them to the generators $W_{i,j}^{(r)}$ of $\mathcal{W}^{\kappa}(\mathfrak{g},f)$.
This gives a description of $\Delta_{l_1,l_2}$ in terms of generators.
We give its explicit form for $r=1,2$ as follows:
\[
	\Delta_{l_1,l_2}(W_{i,j}^{(1)}(m)) = W_{i,j}^{(1)}(m) \otimes 1 + 1 \otimes W_{i,j}^{(1)}(m),
\]
\begin{equation*}
	\begin{split}
		\Delta_{l_1,l_2}(W_{i,j}^{(2)}(m)) &= W_{i,j}^{(2)}(m) \otimes 1 + 1 \otimes W_{i,j}^{(2)}(m) + \sum_{r \in \mathbb{Z}} \sum_{a=1}^{n} W_{i,a}^{(1)}(m+r) \otimes W_{a,j}^{(1)}(-r) \\
		&\qquad - (m+1) l_2 \alpha W_{i,j}^{(1)}(m) \otimes 1.
	\end{split}
\end{equation*}

We will relate $\Delta_{l_1,l_2}$ to the coproduct of the affine Yangian in Corollary~\ref{cor:parabolic_induction} and \ref{cor:parabolic_induction2} as a consequence of the main result of this paper.

\section{Affine Yangians}\label{section:Affine_Yangians}

Assume $n \geq 3$ until the end of Section~\ref{section:Commuting_elements}.
Let
\[
	a_{ij} =
	\begin{cases}
		2  &\text{if } i=j, \\
		-1 &\text{if } i=j \pm 1 \mod n, \\
		0  &\text{otherwise}
	\end{cases}
\]
be entries of the Cartan matrix of type $A_{n-1}^{(1)}$.
We use the notation $\{x,y\}=xy+yx$.

\begin{dfn}\label{dfn:affine_Yangian}
The affine Yangian $Y(\hat{\mathfrak{sl}}_n) = Y_{\hbar,\varepsilon}(\hat{\mathfrak{sl}}_n)$ is the algebra over $\mathbb{C}$ generated by $X_{i,r}^{+}$, $X_{i,r}^{-}$, $H_{i,r}$ ($i = 0,1,\ldots,n-1$ and $r \in \mathbb{Z}_{\geq 0})$ with parameters $\hbar, \varepsilon \in \mathbb{C}$ subject to the relations:
\begin{equation*}
	[H_{i,r},H_{j,s}]=0, \qquad [X_{i,r}^{+},X_{j,s}^{-}] = \delta_{ij} H_{i,r+s}, \qquad [H_{i,0},X_{j,s}^{\pm}]=\pm a_{ij} X_{j,s}^{\pm} \quad \text{ for any $i,j$,}
\end{equation*}
\begin{align*}
	&[H_{i, r+1}, X_{j, s}^{\pm}] - [H_{i, r}, X_{j, s+1}^{\pm}]
	= \pm \dfrac{a_{ij}}{2} \hbar \{ H_{i, r}, X_{j, s}^{\pm} \}, \\
	&[X_{i, r+1}^{\pm}, X_{j, s}^{\pm}] - [X_{i, r}^{\pm}, X_{j, s+1}^{\pm}]
 	= \pm \dfrac{a_{ij}}{2} \hbar \{ X_{i, r}^{\pm}, X_{j, s}^{\pm} \} \quad \text{for $(i,j) \neq (0,n-1), (n-1,0)$,}
\end{align*}
\begin{equation*}
	[H_{n-1, r+1}, X_{0, s}^{\pm}] - [H_{n-1, r}, X_{0, s+1}^{\pm}]
	=\mp\dfrac{\hbar}{2}\{H_{n-1, r}, X_{0, s}^{\pm}\}+\left( \dfrac{n}{2}\hbar + \varepsilon \right)[H_{n-1, r}, X_{0, s}^{\pm}],
\end{equation*}
\begin{equation*}
	[H_{0, r+1}, X_{n-1, s}^{\pm}] - [H_{0, r}, X_{n-1, s+1}^{\pm}]
	=\mp\dfrac{\hbar}{2}\{H_{0, r}, X_{n-1, s}^{\pm}\}-\left( \dfrac{n}{2}\hbar + \varepsilon \right)[H_{0, r}, X_{n-1, s}^{\pm}],
\end{equation*}
\begin{equation*}
	[X_{n-1, r+1}^{\pm}, X_{0, s}^{\pm}] - [X_{n-1, r}^{\pm}, X_{0, s+1}^{\pm}]
	=\mp\dfrac{\hbar}{2}\{X_{n-1, r}^{\pm}, X_{0, s}^{\pm}\}+\left( \dfrac{n}{2}\hbar + \varepsilon \right)[X_{n-1, r}^{\pm}, X_{0, s}^{\pm}],
\end{equation*}
\begin{equation*}
	\sum_{w \in \mathfrak{S}_{1-a_{ij}}} [X_{i,r_{w(1)}}^{\pm}, [X_{i,r_{w(2)}}^{\pm}, \dots, [X_{i,r_{w(1 - a_{ij})}}^{\pm}, X_{j,s}^{\pm}]\dots]] = 0 \quad \text{for $i \neq j$.}
\end{equation*}
\end{dfn}

Set
\[
	X_{i}^{\pm}(z) = \hbar \sum_{r \geq 0} X_{i,r}^{\pm} z^{-r-1}, \quad H_{i}(z) = 1 + \hbar \sum_{r \geq 0} H_{i,r} z^{-r-1}.
\]

\begin{rem}
\begin{enumerate}
\item
The defining relations given in Definition~\ref{dfn:affine_Yangian} is different from those in \cite{MR4207398}.
Generators in \cite{MR4207398} are denoted by $x_{i,r}^{+}$, $x_{i,r}^{-}$, $h_{i,r}$ ($i = 0,1,\ldots,n-1$ and $r \in \mathbb{Z}_{\geq 0})$ with parameters $\varepsilon_1, \varepsilon_2$.
We set $\hbar = \varepsilon_1 + \varepsilon_2$ and
\[
	x_{i}^{\pm}(z) = \hbar \sum_{r \geq 0} x_{i,r}^{\pm} z^{-r-1}, \quad h_{i}(z) = 1 + \hbar \sum_{r \geq 0} h_{i,r} z^{-r-1}.
\]
Then the isomorphism is given by
\[
	X_{i}^{\pm}(z) = x_{i}^{\pm}\left(z-\dfrac{i}{2}( \varepsilon_1-\varepsilon_2 )\right), \quad
	H_{i}(z) = h_{i}\left(z-\dfrac{i}{2}( \varepsilon_1-\varepsilon_2 )\right)
\]
for $i=0,1,\ldots,n-1$ and
\[
	\hbar = \varepsilon_1 + \varepsilon_2, \quad \varepsilon = -n\varepsilon_2.
\]
The condition among the parameters and the central element for the evaluation map in \cite[Theorem~3.8]{MR4207398} is $\hbar c = - n \varepsilon_1$.
This is equivalent to $\hbar(c + n) = - \varepsilon$ in terms of the parameters in this paper.

\item
The presentation in Definition~\ref{dfn:affine_Yangian} is given by Feigin-Finkelberg-Negut-Rybnikov \cite[3.15]{MR2827177}.
They use parameters $\hbar$, $\hbar'$.
The relation to our parameters is:
\[
	\text{$\hbar$ is the same}, \quad \hbar' = -n\varepsilon_1 = -(n\hbar + \varepsilon).
\]
\end{enumerate}
\end{rem}

The following is an immediate consequence of the defining relations.
\begin{prop}\label{prop:degree_one_generator}
The affine Yangian $Y(\hat{\mathfrak{sl}}_n)$ is generated by $X_{i,0}^{+}$, $X_{i,0}^{-}$, and $X_{i,1}^{+}$ for $i=0,1,\ldots,n-1$.
\end{prop}

We define a grading of $Y(\hat{\mathfrak{sl}}_n)$ by setting the degree of $X_{0,r}^{\pm}$ to be $\pm 1$ and that of other generators to be $0$.
Then we denote by $\affY_{\mathrm{comp}}$ the standard degree-wise completion of $Y(\hat{\mathfrak{sl}}_n)$.
Moreover $\affY_{\mathrm{comp}}^{\otimes l}$ denotes the degree-wise completed tensor product defined in the same way.
The target space of the coproduct will be $\affY_{\mathrm{comp}}^{\otimes 2}$.

\section{Coproduct}\label{section:Coproduct}

We introduce the coproduct for $Y(\hat{\mathfrak{sl}}_n)$ according to \cite{MR3861718}.
In \cite{MR3861718}, Guay-Nakajima-Wendlandt constructed the coproduct for affine Yangians, whose explicit form was first discovered by Guay~\cite{MR2323534} for type A.

The affine Lie algebra $\hat{\mathfrak{sl}}_n$ is defined by
\[
	\hat{\mathfrak{sl}}_{n} = \mathfrak{sl}_{n}[t,t^{-1}] \oplus \mathbb{C} c,
\]
\[
	[X t^{m}, Y t^{m'}] = [X,Y] t^{m+m'} + m \delta_{m+m',0} \tr(XY) c, \quad \text{$c$ is central}.
\]
We prepare some notations for $\hat{\mathfrak{sl}}_n$.
We denote the element $X t^{m}$ by $X(m)$.
Set
\begin{gather*}
	X_{0}^{+} = E_{n,1}(1), \quad X_{0}^{-} = E_{1,n}(-1), \quad H_{0} = E_{n,n} - E_{1,1} + c,\\
	X_{i}^{+} = E_{i,i+1}, \quad X_{i}^{-} = E_{i+1,i}, \quad H_{i} = E_{i,i}-E_{i+1,i+1} \ \ (i \neq 0).
\end{gather*}
Let $\hat{\Delta}_{+}$ and $\hat{\Delta}_{+}^{\mathrm{re}}$ be the set of positive roots and of positive real roots.
Let $\mult \gamma$ denote the multiplicity of a root $\gamma$.
We fix root vectors $X_{\pm \gamma}^{(r)}$ for each $\gamma \in \hat{\Delta}_{+}$ and $r=1,\ldots,\mult \gamma$ satisfying $(X_{\gamma}^{(r)},X_{-\gamma}^{(s)})=\delta_{rs}$ with respect to the standard invariant symmetric bilinear form.
The simple roots are denoted by $\{ \alpha_i \mid i=0,1,\ldots,n-1\}$.

The subalgebra of $Y(\hat{\mathfrak{sl}}_n)$ generated by $X_{i,0}^{+}$, $X_{i,0}^{-}$, $H_{i,0}$ ($i = 0,1,\ldots,n-1$) is isomorphic to $U(\hat{\mathfrak{sl}}_n)$ (see \cite[Theorem~6.1]{MR2323534} for $n \geq 4$ and \cite[Theorem~6.9]{MR4014633} in general).
We identify $U(\hat{\mathfrak{sl}}_n)$ with the subalgebra and use the same symbols for corresponding elements of both $U(\hat{\mathfrak{sl}}_n)$ and $Y(\hat{\mathfrak{sl}}_n)$.

We use the symbol $\square$ for $\square(X) = X \otimes 1 + 1 \otimes X$.

\begin{thm}[\cite{MR2323534}, Section~6, pp.\ 462, \cite{MR3861718}, Definition~4.6, Theorem~4.9, Proposition~5.18, Section~7, Proposition~4.24]
There exists an algebra homomorphism $\Delta \colon Y(\hat{\mathfrak{sl}}_n) \to \affY_{\mathrm{comp}}^{\otimes 2}$ uniquely determined by
\[
	\Delta(X_{i,0}^{\pm}) = \square(X_{i,0}^{\pm}),\quad \Delta(H_{i,0}) = \square(H_{i,0}),
\]
\begin{align*}
	\Delta(X_{i,1}^{+}) &= \square(X_{i,1}^{+}) + \hbar \Bigg( X_{i}^{+} \otimes H_{i} - \sum_{\gamma \in \hat{\Delta}_{+}} \sum_{r=1}^{\mult \gamma} [X_{i}^{+}, X_{\gamma}^{(r)}] \otimes X_{-\gamma}^{(r)} \Bigg),\\
	\Delta(X_{i,1}^{-}) &= \square(X_{i,1}^{-}) + \hbar \Bigg( H_{i} \otimes X_{i}^{-} + \sum_{\gamma \in \hat{\Delta}_{+}} \sum_{r=1}^{\mult \gamma} X_{\gamma}^{(r)} \otimes [X_{i}^{-}, X_{-\gamma}^{(r)}] \Bigg),\\
	\Delta(H_{i,1}) &= \square(H_{i,1}) + \hbar \Bigg( H_{i} \otimes H_{i} - \sum_{\gamma \in \hat{\Delta}_{+}^{\mathrm{re}}} (\alpha_i,\gamma)\, X_{\gamma}^{(1)} \otimes X_{-\gamma}^{(1)} \Bigg)
\end{align*}
for $i=0,1,\ldots,n-1$.
Moreover, it satisfies the coassociativity $(\Delta \otimes \id) \circ \Delta = (\id \otimes \Delta) \circ \Delta$.
\end{thm}

We remark that the formulation is different from one in $\cite{MR3861718}$ for two points.
First, in $\cite{MR3861718}$, a completion is defined for Yangians associated with symmetrizable Kac-Moody Lie algebras while our completion is specific to affine type.
We need it as we will consider the composition of the coproduct and the evaluation map, and compare its image with the rectangular $W$-algebra.
Second, the coproduct in this paper is opposite to that of \cite{MR2323534} and \cite{MR3861718}.
Namely the order of the tensor product is reversed.

An explicit form of $\Delta$ is derived by a straightforward computation as follows.

\begin{prop}\label{prop:explicit_coproduct}
We have
\begin{align*}
	&\Delta(X_{0,1}^{+}) = \square(X_{0,1}^{+}) \\
	&\quad + \hbar \Bigg( X_{0}^{+} \otimes c + \sum_{m \geq 0} \sum_{a=1}^{n} \Big( E_{a,1}(m+1) \otimes E_{n,a}(-m) - E_{n,a}(m+1) \otimes E_{a,1}(-m) \Big)\Bigg),\\
	&\Delta(X_{0,1}^{-}) = \square(X_{0,1}^{-}) \\
	&\quad + \hbar \Bigg( c \otimes X_{0}^{-} + \sum_{m \geq 0} \sum_{a=1}^{n} \Big( E_{a,n}(m) \otimes E_{1,a}(-m-1) - E_{1,a}(m) \otimes E_{a,n}(-m-1) \Big)\Bigg)
\end{align*}
and
\begin{align*}
	&\Delta(X_{i,1}^{+}) = \square(X_{i,1}^{+})\\
	&\quad + \hbar \sum_{m \geq 0} \Bigg( \sum_{a=1}^{i} E_{a,i+1}(m) \otimes E_{i,a}(-m) + \sum_{a=i+1}^{n} E_{a,i+1}(m+1) \otimes E_{i,a}(-m-1) \\
	&\qquad\qquad\qquad - \sum_{a=1}^{i} E_{i,a}(m+1) \otimes E_{a,i+1}(-m-1) - \sum_{a=i+1}^{n} E_{i,a}(m) \otimes E_{a,i+1}(-m) \Bigg),\\
	&\Delta(X_{i,1}^{-}) = \square(X_{i,1}^{-})\\
	&\quad + \hbar \sum_{m \geq 0} \Bigg( \sum_{a=1}^{i} E_{a,i}(m) \otimes E_{i+1,a}(-m) + \sum_{a=i+1}^{n} E_{a,i}(m+1) \otimes E_{i+1,a}(-m-1) \\
	&\qquad\qquad\qquad - \sum_{a=1}^{i} E_{i+1,a}(m+1) \otimes E_{a,i}(-m-1) - \sum_{a=i+1}^{n} E_{i+1,a}(m) \otimes E_{a,i}(-m) \Bigg)
\end{align*}
for $i=1,\ldots,n-1$.
\end{prop}
We do not use an explicit form of $\Delta(H_{i,1})$.

\section{Evaluation map}\label{section:Evaluation_map}

The evaluation map for $Y(\hat{\mathfrak{sl}}_n)$ was discovered by Guay in \cite{MR2323534}.
It requires a certain condition among the parameters as remarked in \cite{MR4207398}.
We state it in the form for the presentation of this paper.

\begin{rem}\label{rem:error}
The published version of \cite{MR4207398} has an error and the correction is available.
See also the arXiv version of \cite{MR4207398}, or Section~\ref{section:Commuting_elements} and Appendix~\ref{section:proof_of_evaluation_map} of this paper.
\end{rem}

Recall that we set $\alpha = k + (l-1)n$ and defined $\hat{\mathfrak{gl}}_{n}^{\alpha}$ as the direct sum of $\hat{\mathfrak{sl}}_{n}$ at level $\alpha$ and the Heisenberg Lie algebra at level $\alpha+n$ in Section~\ref{section:Affine_Lie_algebras}.
Although Theorem~\ref{thm:evaluation} below holds for an arbitrary $\alpha$ satisfying $\alpha + n = -\varepsilon/\hbar$, we use it with $\alpha = k + (l-1)n$ for our purpose.

\begin{thm}[\cite{MR2323534}, Section~6, pp.\ 462--463, \cite{MR4207398}, Theorem~3.8]\label{thm:evaluation}
Assume $\alpha + n = -\varepsilon/\hbar$.
Then there exists an algebra homomorphism $\ev \colon Y(\hat{\mathfrak{sl}}_n) \to U(\hat{\mathfrak{gl}}_{n}^{\alpha})_{{\mathrm{comp}}}$ uniquely determined by
\[
	\ev(X_{i,0}^{\pm})=X_{i}^{\pm}, \quad \ev(H_{i,0})=H_{i}, \quad \ev(c) = \alpha,
\]
\begin{align*}
		\ev(X_{0,1}^{+}) &= \hbar \Bigg( \alpha X_{0}^{+} + \sum_{m \geq 0} \sum_{a=1}^{n} E_{n,a}(-m) E_{a,1}(m+1) \Bigg),\\
		\ev(X_{0,1}^{-}) &= \hbar \Bigg( \alpha X_{0}^{-} + \sum_{m \geq 0} \sum_{a=1}^{n} E_{1,a}(-m-1) E_{a,n}(m) \Bigg),\\
		\ev(H_{0,1}) &= \hbar \Bigg( \alpha H_{0} - E_{n,n} (E_{1,1}-\alpha) + \sum_{m \geq 0} \sum_{a=1}^{n} \Big( E_{n,a}(-m) E_{a,n}(m) - E_{1,a}(-m-1) E_{a,1}(m+1) \Big)\Bigg)
\end{align*}
and
\begin{align*}
		\ev(X_{i,1}^{+}) &= \hbar \Bigg( -\frac{i}{2} X_{i}^{+} + \sum_{m \geq 0} \Bigg( \sum_{a=1}^{i} E_{i,a}(-m) E_{a,i+1}(m) + \sum_{a=i+1}^{n} E_{i,a}(-m-1) E_{a,i+1}(m+1) \Bigg) \Bigg),\\
		\ev(X_{i,1}^{-}) &= \hbar \Bigg( -\frac{i}{2} X_{i}^{-} + \sum_{m \geq 0} \Bigg( \sum_{a=1}^{i} E_{i+1,a}(-m) E_{a,i}(m) + \sum_{a=i+1}^{n} E_{i+1,a}(-m-1) E_{a,i}(m+1) \Bigg) \Bigg),\\
		\ev(H_{i,1}) &= \hbar \Bigg( -\frac{i}{2} H_{i} - E_{i,i} E_{i+1,i+1} + \sum_{m \geq 0} \Bigg( \sum_{a=1}^{i} E_{i,a}(-m) E_{a,i}(m) + \sum_{a=i+1}^{n} E_{i,a}(-m-1) E_{a,i}(m+1) \\
		&- \sum_{a=1}^{i} E_{i+1,a}(-m) E_{a,i+1}(m) - \displaystyle\sum_{a=i+1}^{n} E_{i+1,a}(-m-1) E_{a,i+1}(m+1) \Bigg) \Bigg)
\end{align*}
for $i=1,\ldots,n-1$.
\end{thm}

\section{Affine Yangians and rectangular $W$-algebras}\label{section:Affine_Yangians_and_rectangular_W-algebras}

For each $\beta \in \mathbb{C}$, we define an algebra automorphism $\eta_{\beta}$ of $U(\hat{\mathfrak{gl}}_{n}^{\alpha})$ by
\[
	E_{i,j}(m) \mapsto E_{i,j}(m) + \delta_{m,0}\delta_{i,j}\beta.
\]
We define an algebra automorphism $\eta^{(l)}$ of $U(\hat{\mathfrak{gl}}_{n}^{\alpha})_{{\mathrm{comp}}}^{\otimes l}$ by
\[
	\eta^{(l)} = \eta_{(l-1)\alpha} \otimes \eta_{(l-2)\alpha} \otimes \cdots \otimes \eta_{\alpha} \otimes \id.
\]

We denote by $\Delta^{l-1}$ the $(l-1)$ times iterated composition of $\Delta$, which is a homomorphism from $\affY_{\mathrm{comp}}$ to $\affY_{\mathrm{comp}}^{\otimes l}$.
The map $\ev^{\otimes l} \colon Y(\hat{\mathfrak{sl}}_n)^{\otimes l} \to U(\hat{\mathfrak{gl}}_{n}^{\alpha})_{{\mathrm{comp}}}^{\otimes l}$ is extended to an algebra homomorphism from $\affY_{\mathrm{comp}}^{\otimes l}$ to $U(\hat{\mathfrak{gl}}_{n}^{\alpha})_{{\mathrm{comp}}}^{\otimes l}$.
We use the same letter $\ev^{\otimes l}$ for this map.

\begin{dfn}\label{dfn:map}
Assume $k+N=\alpha+n=-\varepsilon/\hbar$.
We set
\[
	\Phi_{l} = \eta^{(l)} \circ \ev^{\otimes l} \circ \Delta^{l-1},
\]
which is an algebra homomorphism from $Y(\hat{\mathfrak{sl}}_n)_{\mathrm{comp}}$ to $U(\hat{\mathfrak{gl}}_{n}^{\alpha})_{{\mathrm{comp}}}^{\otimes l}$.
\end{dfn}
Let us state the main theorem.
\begin{thm}\label{thm:main}
The image of $\Phi_{l}$ is contained in $\mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f))$.
That is, the map $\Phi_{l} = \eta^{(l)} \circ \ev^{\otimes l} \circ \Delta^{l-1}$ gives an algebra homomorphism
\[
	\Phi_{l} \colon Y(\hat{\mathfrak{sl}}_n)_{\mathrm{comp}} \to \mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f)).
\]
Moreover, it is surjective if $\alpha \neq 0$.
\end{thm}

\begin{proof}
Since we have
\begin{equation}
		\Phi_{l}(X_{i,0}^{+}) = \begin{cases}
			W_{n,1}^{(1)}(1) & \text{ if $i=0$},\\
			W_{i,i+1}^{(1)}(0) & \text{ if $i \neq 0$},
		\end{cases}\quad
		\Phi_{l}(X_{i,0}^{-}) = \begin{cases}
			W_{1,n}^{(1)}(-1) & \text{ if $i=0$},\\
			W_{i+1,i}^{(1)}(0) & \text{ if $i \neq 0$},
		\end{cases}\label{eq:image_of_degree_zero}
\end{equation}
the image of $\{ X_{i,0}^{\pm} \mid 0 \leq i \leq n-1 \}$ is contained in $\mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f))$.
Hence the following formulas conclude that the image of $\Phi_{l}$ is contained in $\mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f))$ by Proposition~\ref{prop:degree_one_generator}.
\begin{prop}\label{prop:image}
We have
\begin{equation*}
	\begin{split}
		\Phi_{l}(X_{0,1}^{+}) &= (-\hbar) \Bigg( W_{n,1}^{(2)}(1) - \alpha W_{n,1}^{(1)}(1) - \sum_{m \geq 0} \sum_{a=1}^{n} W_{n,a}^{(1)}(-m) W_{a,1}^{(1)}(m+1)\Bigg),\\
		\Phi_{l}(X_{0,1}^{-}) &= (-\hbar) \Bigg( W_{1,n}^{(2)}(-1) - l \alpha W_{1,n}^{(1)}(-1) - \sum_{m \geq 0} \sum_{a=1}^{n} W_{1,a}^{(1)}(-m-1) W_{a,n}^{(1)}(m) \Bigg)
	\end{split}
\end{equation*}
and
\begin{equation*}
	\begin{split}
		\Phi_{l}(X_{i,1}^{+}) &= (-\hbar) \Bigg( W_{i,i+1}^{(2)}(0) + \dfrac{i}{2} W_{i,i+1}^{(1)}(0) \\
		&\qquad - \sum_{m \geq 0} \Bigg( \sum_{a=1}^{i} W_{i,a}^{(1)}(-m) W_{a,i+1}^{(1)}(m) + \sum_{a=i+1}^{n} W_{i,a}^{(1)}(-m-1) W_{a,i+1}^{(1)}(m+1) \Bigg)\Bigg),\\
		\Phi_{l}(X_{i,1}^{-}) &= (-\hbar) \Bigg( W_{i+1,i}^{(2)}(0) + \dfrac{i}{2} W_{i+1,i}^{(1)}(0) \\
		&\qquad - \sum_{m \geq 0} \Bigg( \sum_{a=1}^{i} W_{i+1,a}^{(1)}(-m) W_{a,i}^{(1)}(m) + \sum_{a=i+1}^{n} W_{i+1,a}^{(1)}(-m-1) W_{a,i}^{(1)}(m+1) \Bigg)\Bigg)
	\end{split}
\end{equation*}
for $i=1,\ldots,n-1$.
\end{prop}

\begin{proof}[Proof of Proposition~\ref{prop:image}]
We show the assertions only for $i = 0$.
Note that we have an equality $\square^{l-1}(X) = \sum_{s=1}^{l} X^{[s]}$ in $U(\hat{\mathfrak{gl}}_{n}^{\alpha})_{{\mathrm{comp}}}^{\otimes l}$.

We show the assertion for $X_{0,1}^{+}$.
We have
\begin{equation*}
	\begin{split}
		&\ev^{\otimes l} \circ \Delta^{l-1}(X_{0,1}^{+}) = \square^{l-1}(\ev(X_{0,1}^{+})) \\
		&\ +\hbar \sum_{1 \leq s_1 < s_2 \leq l} \ev^{\otimes l} \Bigg( E_{n,1}^{[s_1]}(1) (c^{[s_2]}) + \sum_{m \geq 0} \sum_{a=1}^{n} \Big( E_{a,1}^{[s_1]}(m+1) E_{n,a}^{[s_2]}(-m) - E_{n,a}^{[s_1]}(m+1) E_{a,1}^{[s_2]}(-m) \Big)\Bigg)\\
		&= (-\hbar) \Bigg( - \alpha \sum_{s=1}^{l} E_{n,1}^{[s]}(1) - \sum_{m \geq 0} \sum_{s=1}^{l} \sum_{a=1}^{n} E_{n,a}^{[s]}(-m) E_{a,1}^{[s]}(m+1) - \alpha \sum_{s=1}^{l} (l-s) E_{n,1}^{[s]}(1) \\
		&\qquad\qquad\qquad\qquad - \sum_{m \geq 0} \sum_{1 \leq s_1 < s_2 \leq l} \sum_{a=1}^{n} \Big( E_{a,1}^{[s_1]}(m+1) E_{n,a}^{[s_2]}(-m) - E_{n,a}^{[s_1]}(m+1) E_{a,1}^{[s_2]}(-m) \Big)\Bigg).
	\end{split}
\end{equation*}
This is equal to
\begin{equation*}
	\begin{split}
		&(-\hbar) \Bigg( - \alpha \sum_{s=1}^{l} E_{n,1}^{[s]}(1) - \alpha \sum_{s=1}^{l} (l-s) E_{n,1}^{[s]}(1) + \sum_{m \in \mathbb{Z}} \sum_{1 \leq s_1 < s_2 \leq l} \sum_{a=1}^{n} E_{n,a}^{[s_1]}(m+1) E_{a,1}^{[s_2]}(-m) \\
		&\qquad\qquad - \sum_{m \geq 0} \sum_{a=1}^{n} \left( \sum_{s_1=1}^{l} E_{n,a}^{[s_1]}(-m) \right) \left( \sum_{s_2=1}^{l} E_{a,1}^{[s_2]}(m+1) \right) \Bigg).
	\end{split}
\end{equation*}
Then we obtain
\begin{equation}
	\begin{split}
		&\eta^{(l)} \circ \ev^{\otimes l} \circ \Delta^{l-1}(X_{0,1}^{+}) = \ev^{\otimes l} \circ \Delta^{l-1}(X_{0,1}^{+}) \\
		&\quad + (-\hbar) \Bigg( \sum_{1 \leq s_1 < s_2 \leq l} \Big( (l-s_2)\alpha E_{n,1}^{[s_1]}(1) + (l-s_1)\alpha E_{n,1}^{[s_2]}(1) \Big) - \sum_{s_1=1}^{l} (l-s_1)\alpha \sum_{s_2=1}^{l} E_{n,1}^{[s_2]}(1) \Bigg)\\
		&= (-\hbar) \Bigg( - \alpha \sum_{s=1}^{l} E_{n,1}^{[s]}(1) - \alpha \sum_{s=1}^{l} (l-s) E_{n,1}^{[s]}(1) + \sum_{m \in \mathbb{Z}} \sum_{a=1}^{n} \sum_{1 \leq s_1 < s_2 \leq l} E_{n,a}^{[s_1]}(m+1) E_{a,1}^{[s_2]}(-m) \\
		&\qquad\qquad - \sum_{m \geq 0} \sum_{a=1}^{n} \left( \sum_{s_1=1}^{l} E_{n,a}^{[s_1]}(-m) \right) \left( \sum_{s_2=1}^{l} E_{a,1}^{[s_2]}(m+1) \right) - \alpha \sum_{s=1}^{l} (l-s) E_{n,1}^{[s]}(1) \Bigg) \label{eq:Phi}.
	\end{split}
\end{equation}
Since we have
\begin{equation*}
	\begin{split}
		W_{i,j}^{(1)}(m) &= \sum_{s=1}^{l} E_{i,j}^{[s]}(m),\\
		W_{n,1}^{(2)}(1) &= \sum_{m \in \mathbb{Z}} \sum_{1 \leq s_1 < s_2 \leq l} \sum_{a=1}^{n} E_{n,a}^{[s_1]}(m+1) E_{a,1}^{[s_2]}(-m) - 2\alpha \sum_{s=1}^{l} (l-s) E_{n,1}^{[s]}(1)
	\end{split}
\end{equation*}
by Example~\ref{exam:generators}, we see that (\ref{eq:Phi}) is equal to
\[
	(-\hbar) \Bigg( W_{n,1}^{(2)}(1) - \alpha W_{n,1}^{(1)}(1) - \sum_{m \geq 0} \sum_{a=1}^{n} W_{n,a}^{(1)}(-m) W_{a,1}^{(1)}(m+1)\Bigg).
\]

We show the assertion for $X_{0,1}^{-}$.
We have
\begin{equation*}
	\begin{split}
		&\ev^{\otimes l} \circ \Delta^{l-1}(X_{0,1}^{-}) = \square^{l-1}(\ev(X_{0,1}^{-})) \\
		&\ + \hbar \sum_{1 \leq s_1 < s_2 \leq l} \ev^{\otimes l} \Bigg( (c^{[s_1]}) E_{1,n}^{[s_2]}(-1) + \sum_{m \geq 0} \sum_{a=1}^{n} \Big( E_{a,n}^{[s_1]}(m) E_{1,a}^{[s_2]}(-m-1) - E_{1,a}^{[s_1]}(m) E_{a,n}^{[s_2]}(-m-1) \Big)\Bigg)\\
		&= (-\hbar) \Bigg( - \alpha \sum_{s=1}^{l} E_{1,n}^{[s]}(-1) - \sum_{m \geq 0} \sum_{s=1}^{l} \sum_{a=1}^{n} E_{1,a}^{[s]}(-m-1) E_{a,n}^{[s]}(m) - \alpha \sum_{s=1}^{l} (s-1) E_{1,n}^{[s]}(-1) \\
		&\qquad\qquad\qquad\qquad - \sum_{m \geq 0} \sum_{1 \leq s_1 < s_2 \leq l} \sum_{a=1}^{n} \Big( E_{a,n}^{[s_1]}(m) E_{1,a}^{[s_2]}(-m-1) - E_{1,a}^{[s_1]}(m) E_{a,n}^{[s_2]}(-m-1) \Big)\Bigg).
	\end{split}
\end{equation*}
This is equal to
\begin{equation*}
	\begin{split}
		&(-\hbar) \Bigg( - l \alpha \sum_{s=1}^{l} E_{1,n}^{[s]}(-1) + \alpha \sum_{s=1}^{l} (l-s) E_{1,n}^{[s]}(-1) + \sum_{m \in \mathbb{Z}} \sum_{1 \leq s_1 < s_2 \leq l} \sum_{a=1}^{n} E_{1,a}^{[s_1]}(m) E_{a,n}^{[s_2]}(-m-1) \\
		&\qquad\qquad - \sum_{m \geq 0} \sum_{a=1}^{n} \left( \sum_{s_1=1}^{l} E_{1,a}^{[s_1]}(-m-1) \right) \left( \sum_{s_2=1}^{l} E_{a,n}^{[s_2]}(m) \right) \Bigg).
	\end{split}
\end{equation*}
Then we obtain
\begin{equation}
	\begin{split}
		&\eta^{(l)} \circ \ev^{\otimes l} \circ \Delta^{l-1}(X_{0,1}^{-}) = \ev^{\otimes l} \circ \Delta^{l-1}(X_{0,1}^{-}) \\
		&\quad + (-\hbar) \Bigg( \sum_{1 \leq s_1 < s_2 \leq l} \Big( (l-s_2)\alpha E_{1,n}^{[s_1]}(-1) + (l-s_1)\alpha E_{1,n}^{[s_2]}(-1) \Big) - \sum_{s_2=1}^{l} (l-s_2)\alpha \sum_{s_1=1}^{l} E_{1,n}^{[s_1]}(-1) \Bigg)\\
		&= (-\hbar) \Bigg( - l \alpha \sum_{s=1}^{l} E_{1,n}^{[s]}(-1) + \alpha \sum_{s=1}^{l} (l-s) E_{1,n}^{[s]}(-1) + \sum_{m \in \mathbb{Z}} \sum_{a=1}^{n} \sum_{1 \leq s_1 < s_2 \leq l} E_{1,a}^{[s_1]}(m) E_{a,n}^{[s_2]}(-m-1) \\
		&\qquad\qquad\quad - \sum_{m \geq 0} \sum_{a=1}^{n} \left( \sum_{s_1=1}^{l} E_{1,a}^{[s_1]}(-m-1) \right) \left( \sum_{s_2=1}^{l} E_{a,n}^{[s_2]}(m) \right) - \alpha \sum_{s=1}^{l} (l-s) E_{1,n}^{[s]}(-1) \Bigg) \label{eq:Phi2}.
	\end{split}
\end{equation}
Since we have
\begin{equation*}
		W_{1,n}^{(2)}(-1) = \sum_{m \in \mathbb{Z}} \sum_{1 \leq s_1 < s_2 \leq l} \sum_{a=1}^{n} E_{1,a}^{[s_1]}(m) E_{a,n}^{[s_2]}(-m-1)
\end{equation*}
by Example~\ref{exam:generators}, we see that (\ref{eq:Phi2}) is equal to
\[
	(-\hbar) \Bigg( W_{1,n}^{(2)}(-1) - l \alpha W_{1,n}^{(1)}(-1) - \sum_{m \geq 0} \sum_{a=1}^{n} W_{1,a}^{(1)}(-m-1) W_{a,n}^{(1)}(m) \Bigg).
\]
\end{proof}
\noindent \textbf{Continuation of the proof of Theorem~\ref{thm:main}}:
The map $\Phi_{l}$ coincides with $\Phi$ constructed by the second named author in \cite{Affine_super_Yangians_and_rectangular_W-superalgebras} up to the twist described in Remark~\ref{rem:difference} (i).
Therefore the surjectivity of $\Phi_{l}$ follows from \cite{Affine_super_Yangians_and_rectangular_W-superalgebras}.
We give some part of the discussion in Appendix~\ref{section:surjectivity} for completeness.
\end{proof}

\begin{cor}
Let $M= \bigoplus_{\gamma \in \mathbb{C}} M_{\gamma}$ be a graded $\mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f))$-module and for each $\gamma$ we assume that $M_{\gamma + d} = 0$ holds for sufficiently large $d \in \mathbb{Z}$.
Then $M$ is regarded as a $Y(\hat{\mathfrak{sl}}_n)$-module via $\Phi_{l}$.

Moreover, each $M_{\gamma}$ is regarded as a module of the Yangian $Y(\mathfrak{sl}_{n})$ of finite type by restriction.
\end{cor}
\begin{proof}
The former assertion is immediate.
For the latter one, we see that the subalgebra of $Y(\hat{\mathfrak{sl}}_n)$ generated by $X_{i,r}^{\pm}$, $H_{i,r}$ ($i=1,\ldots,n-1$ and $r \in \mathbb{Z}_{\geq 0}$) is isomorphic to the Yangian $Y(\mathfrak{sl}_{n})$ of finite type and lies in the degree zero component.
Thus $\Phi_{l}(Y(\mathfrak{sl}_{n}))$ lies in the degree zero component of $\mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f))$.
This proves the assertion.
\end{proof}

\section{Coproduct and parabolic induction}\label{section:Coproduct_and_parabolic_induction}

Assume $l=l_1+l_2$.
Then $\mathcal{W}^{\kappa_i}(\mathfrak{gl}_{N_i},f_i)$ for $i=1,2$ were defined in Section~\ref{section:Parabolic_induction}.
Let
\[
	\Delta_{l_1,l_2} \colon \mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f)) \to \mathfrak{U}(\mathcal{W}^{\kappa_1}(\mathfrak{gl}_{N_1},f_1)) \otimes_{\mathrm{comp}} \mathfrak{U}(\mathcal{W}^{\kappa_2}(\mathfrak{gl}_{N_2},f_2))
\]
be the injective algebra homomorphism induced from the identity map on $U(\hat{\mathfrak{gl}}_{n}^{\alpha})_{\mathrm{comp}}^{\otimes l}$.
Here $\otimes_{\mathrm{comp}}$ denotes the degree-wise completed tensor product defined in the same way as one in Section~\ref{section:Affine_Lie_algebras}.
We give a compatibility of the coproduct $\Delta$ for $Y(\hat{\mathfrak{sl}}_n)$ and $\Delta_{l_1,l_2}$.
Recall the algebra automorphism $\eta_{\beta}$ ($\beta \in \mathbb{C}$) of $U(\hat{\mathfrak{gl}}_{n}^{\alpha})$ defined in Section~\ref{section:Affine_Yangians_and_rectangular_W-algebras}.

\begin{lem}\label{lem:eta}
For any $\beta$, the algebra automorphism $\eta_{\beta}^{\otimes l}$ preserves $\mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f))$ inside $U(\hat{\mathfrak{gl}}_{n}^{\alpha})_{\mathrm{comp}}^{\otimes l}$.
\end{lem}

\begin{proof}
First let us consider the case $\alpha \neq 0$.
Proposition~\ref{prop:generators} asserts that $\mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f))$ is generated by $W_{i,j}^{(r)}(m)$ for $r=1,2$ and all $i,j,m$.
By a direct computation, we have
\begin{align*}
	\eta_{\beta}^{\otimes l}(W_{i,j}^{(1)}(m)) &= W_{i,j}^{(1)}(m) + \delta_{m,0}\delta_{i,j} \beta l, \\
	\eta_{\beta}^{\otimes l}(W_{i,j}^{(2)}(m)) &= W_{i,j}^{(2)}(m) + \beta (l-1) W_{i,j}^{(1)}(m) + \delta_{m,0}\delta_{i,j} \beta(\beta-\alpha) \dfrac{l(l-1)}{2},
\end{align*}
which proves the assertion.

Next consider the case $\alpha = 0$.
In this case, we have
\[
	W_{i,j}^{(r)}(m) = \sum_{\substack{m_1,\ldots,m_r \in \mathbb{Z}\\ m_1+\cdots+m_r=m}}\ \sum_{1 \leq s_1 < \cdots < s_r \leq l}\ \sum_{a_1,\ldots,a_{r-1}=1}^{n} E_{i,a_1}^{[s_1]}(m_1) E_{a_1,a_2}^{[s_2]}(m_2) \cdots E_{a_{r-1},j}^{[s_r]}(m_r)
\]
for $r=1,\ldots,l$, and $\mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f))$ is generated by these elements.
By a direct computation, we have
\[
	\eta_{\beta}^{\otimes l}(W_{i,j}^{(r)}(m)) = \sum_{s=0}^{r} \beta^{s}
	\begin{pmatrix}
		l-r+s\\
		s
	\end{pmatrix} W_{i,j}^{(r-s)}(m),
\]
where $W_{i,j}^{(0)}(m)$ is regarded as $\delta_{m,0}\delta_{i,j}$.
This proves the assertion.
\end{proof}

Let us define $\widetilde{\Delta}_{l_1,l_2}$, a slight modification of $\Delta_{l_1,l_2}$, by
\[
	\widetilde{\Delta}_{l_1,l_2} = (\eta_{-l_2 \alpha}^{\otimes l_1} \otimes \id^{\otimes l_2}) \circ \Delta_{l_1,l_2}.
\]
This gives an injective algebra homomorphism from $\mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f))$ to $\mathfrak{U}(\mathcal{W}^{\kappa_1}(\mathfrak{gl}_{N_1},f_1)) \otimes_{\mathrm{comp}} \mathfrak{U}(\mathcal{W}^{\kappa_2}(\mathfrak{gl}_{N_2},f_2))$ since we have $\eta_{-l_2 \alpha}^{\otimes l_1} \mathfrak{U}(\mathcal{W}^{\kappa_1}(\mathfrak{gl}_{N_1},f_1)) = \mathfrak{U}(\mathcal{W}^{\kappa_1}(\mathfrak{gl}_{N_1},f_1))$ by applying Lemma~\ref{lem:eta}.
\begin{cor}\label{cor:parabolic_induction}
We have
\[
	\widetilde{\Delta}_{l_1,l_2} \circ \Phi_{l} = \left( \Phi_{l_1} \otimes \Phi_{l_2} \right) \circ \Delta.
\]
\end{cor}
\begin{proof}
The assertion is proved by
\begin{equation*}
	\begin{split}
		\widetilde{\Delta}_{l_1,l_2} \circ \Phi_{l} &= (\eta_{-l_2 \alpha}^{\otimes l_1} \otimes \id^{\otimes l_2}) \circ \eta^{(l)} \circ \ev^{\otimes l} \circ \Delta^{l-1}\\
		&= (\eta^{(l_1)} \otimes \eta^{(l_2)}) \circ \ev^{\otimes l} \circ \Delta^{l-1} = \left( \Phi_{l_1} \otimes \Phi_{l_2} \right) \circ \Delta.
	\end{split}
\end{equation*}
\end{proof}

We can reformulate our main theorem and the above compatibility as follows.
Put
\[
	{}^\eta \mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f)) = ( \eta^{(l)} )^{-1} \mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f)), \ \ {}^\eta \mathfrak{U}(\mathcal{W}^{\kappa_i}(\mathfrak{gl}_{N_i},f_i)) = ( \eta^{(l_i)} )^{-1} \mathfrak{U}(\mathcal{W}^{\kappa_i}(\mathfrak{gl}_{N_i},f_i)).
\]
Since we have
\begin{equation*}
	\begin{split}
		( \eta^{(l)} )^{-1} \mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f)) &= (\eta^{(l_1)} \otimes \eta^{(l_2)})^{-1} \circ (\eta_{-l_2 \alpha}^{\otimes l_1} \otimes \id^{\otimes l_2})\, \mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f)) \\
		&\subset (\eta^{(l_1)} \otimes \eta^{(l_2)})^{-1} \circ (\eta_{-l_2 \alpha}^{\otimes l_1} \otimes \id^{\otimes l_2})\, \mathfrak{U}(\mathcal{W}^{\kappa_1}(\mathfrak{gl}_{N_1},f_1)) \otimes_{\mathrm{comp}} \mathfrak{U}(\mathcal{W}^{\kappa_2}(\mathfrak{gl}_{N_2},f_2))\\
		&= (\eta^{(l_1)} \otimes \eta^{(l_2)})^{-1} \mathfrak{U}(\mathcal{W}^{\kappa_1}(\mathfrak{gl}_{N_1},f_1)) \otimes_{\mathrm{comp}} \mathfrak{U}(\mathcal{W}^{\kappa_2}(\mathfrak{gl}_{N_2},f_2)),
	\end{split}
\end{equation*}
we see that ${}^\eta \mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f))$ is contained in ${}^\eta \mathfrak{U}(\mathcal{W}^{\kappa_1}(\mathfrak{gl}_{N_1},f_1)) \otimes_{\mathrm{comp}} {}^\eta \mathfrak{U}(\mathcal{W}^{\kappa_2}(\mathfrak{gl}_{N_2},f_2))$.
We denote by ${}^\eta \Delta_{l_1,l_2}$ this inclusion.
Put
\[
	{}^{\eta} \Phi_{l} = ( \eta^{(l)} )^{-1} \circ \Phi_{l} = \ev^{\otimes l} \circ \Delta^{l-1}.
\]
Now the following is obvious.
\begin{cor}\label{cor:parabolic_induction2}
The map ${}^{\eta} \Phi_{l} = \ev^{\otimes l} \circ \Delta^{l-1}$ gives an algebra homomorphism
\[
	{}^{\eta} \Phi_{l} \colon Y(\hat{\mathfrak{sl}}_n)_{\mathrm{comp}} \to {}^\eta \mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f))
\]
satisfying
\[
	{}^{\eta} \Delta_{l_1,l_2} \circ {}^{\eta} \Phi_{l} = \left( {}^{\eta} \Phi_{l_1} \otimes {}^{\eta} \Phi_{l_2} \right) \circ \Delta.
\]
Moreover, it is surjective if $\alpha \neq 0$.
\end{cor}

\section{Commuting elements}\label{section:Commuting_elements}

By our main result, the elements $\Phi_{l}(H_{i,r})$ for $i= 0, 1, \ldots, n-1$ and $r \in \mathbb{Z}_{\geq 0}$ generate a commutative subalgebra of $\mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f))$.
We expect that the subalgebra will play an important role for applications to the study of integrable systems.
Let us mention a few observation for commuting elements of low degree here.

Put
\begin{equation*}
	D_{i} = W_{i,i}^{(2)}(0) - \sum_{m \geq 0} \left( \sum_{a=1}^{i-1} W_{i,a}^{(1)}(-m) W_{a,i}^{(1)}(m) + \sum_{a=i}^{n} W_{i,a}^{(1)}(-m-1) W_{a,i}^{(1)}(m+1) \right)
\end{equation*}
for $i=1,\ldots,n$.
This element is an analog of $T_{i,i;i-1}^{(2)}$ in \cite[Example~9.1]{MR2199632}
(See also \cite[Section~3.3 (3.10) and Section~3.4]{MR2456464}).
Note that $D_{i}$ is also equal to
\begin{equation*}
	W_{i,i}^{(2)}(0) - \sum_{m \geq 0} \left( \sum_{a=1}^{i} W_{i,a}^{(1)}(-m) W_{a,i}^{(1)}(m) + \sum_{a=i+1}^{n} W_{i,a}^{(1)}(-m-1) W_{a,i}^{(1)}(m+1) \right) + W_{i,i}^{(1)}(0)^2.
\end{equation*}
We can write down $\Phi_{l}(H_{i,1})$ in terms of $D_{i}$'s as:
\begin{equation}
	\begin{split}
		&\Phi_{l}(H_{i,1}) = (-\hbar) \times \\
		&= \begin{cases}
			\Bigg( D_{n} - D_{1} - l\alpha \Big( W_{n,n}^{(1)}(0) - W_{1,1}^{(1)}(0) + l\alpha \Big) - W_{n,n}^{(1)}(0)^2 + W_{n,n}^{(1)}(0) \Big( W_{1,1}^{(1)}(0) - \alpha \Big) \Bigg) & (i = 0),\\
			\Bigg( D_{i} - D_{i+1} + \dfrac{i}{2} \Big( W_{i,i}^{(1)}(0) - W_{i+1,i+1}^{(1)}(0) \Big) - W_{i,i}^{(1)}(0)^2 + W_{i,i}^{(1)}(0) W_{i+1,i+1}^{(1)}(0) \Bigg) & (i \neq 0).
		\end{cases} \label{eq:image_of_H}
	\end{split}
\end{equation}
We note that (\ref{eq:image_of_H}) coincides with the formula for the evaluation map in Theorem~\ref{thm:evaluation} when $l=1$, if we put $W_{i,i}^{(2)}(0) = 0$.

By a lengthy computation, we see that $[D_{i}, D_{j}] = 0$ holds.

\begin{prop}\label{prop:commuting_elements}
We have $[D_{i}, D_{j}] = 0$.
\end{prop}
\begin{proof}
A proof is given in Appendix~\ref{section:commutativity}.
\end{proof}

Moreover we easily see that $[D_{i}, W_{j,j}^{(1)}(0)] = 0$ holds.
Thus these computations imply that the right-hand sides of (\ref{eq:image_of_H}) for any $i,j$ mutually commute without using our main result.
This approach was used for a direct construction of $\Phi_{l}$ by the second named author in \cite{Affine_super_Yangians_and_rectangular_W-superalgebras}.

Let us say a few words on Proposition~\ref{prop:commuting_elements}.
If we put
\[
	A_{i} = \sum_{m \geq 0} \sum_{a=1}^{i-1} W_{i,a}^{(1)}(-m) W_{a,i}^{(1)}(m),\quad B_{i} = \sum_{m \geq 0} \sum_{a=i}^{n} W_{i,a}^{(1)}(-m-1) W_{a,i}^{(1)}(m+1),
\]
then computations for
\[
	[W_{i,i}^{(2)}(0), W_{j,j}^{(2)}(0)], \quad [W_{i,i}^{(2)}(0), A_{j}+B_{j}], \quad [A_{i}+B_{i},A_{j}+B_{j}]
\]
yield the vanishing of $[D_{i}, D_{j}]$.
We have
\begin{equation}
	[A_{i}+B_{i},A_{j}+B_{j}] = (l-1) \sum_{m \geq 1} m \Big( W_{i,i}^{(1)}(-m)W_{j,j}^{(1)}(m) - W_{j,j}^{(1)}(-m)W_{i,i}^{(1)}(m) \Big). \label{eq:AB}
\end{equation}
We write down $[W_{i,i}^{(2)}(0), W_{j,j}^{(2)}(0)]$ and $[W_{i,i}^{(2)}(0), A_{j}+B_{j}]$, and give a proof of Proposition~\ref{prop:commuting_elements} in Appendix~\ref{section:commutativity} for readers' convenience.

In the case $l=1$, the map $\Phi_{1}$ coincides with the evaluation map $\ev$,
and we regard $W_{i,j}^{(2)}(m)$ as $0$.
Then (\ref{eq:AB}) for $l=1$ implies that $[\ev(H_{i,1}), \ev(H_{j,1})]=0$ holds.
This is the main part of computations done by the first named author in \cite{MR4207398} to check that $\ev$ respects the defining relations of $Y(\hat{\mathfrak{sl}}_n)$.
See Appendix~\ref{section:proof_of_evaluation_map} for a derivation of (\ref{eq:AB}).
As mentioned in Remark~\ref{rem:error}, the published version of \cite{MR4207398} has an error.
Although the correction to \cite{MR4207398} is available, we include some explanation in this paper since we use conventions slightly different from \cite{MR4207398}.

\section{Super case}\label{section:Super_case}

Let $m,n$ be nonzero positive integers with $m,n \geq 2$ and $m \neq n$.
Set $M=lm$ and $N=ln$.
For a fixed complex number $k$, we put $\alpha = k+(M-N)-(m-n) = k+(l-1)(m-n)$.

We set $\pari{i}$ to be $0$ if $1 \leq i \leq m$ and to be $1$ if $m+1 \leq i \leq m+n$.
Let $\mathfrak{gl}_{m|n}$ denote the general linear Lie superalgebra with the standard parity.
That is, the parity of the element $E_{i,j}$ ($1 \leq i,j \leq m+n$) is given by $\pari{i}+\pari{j}$.
We define an affinization $\hat{\mathfrak{gl}}_{m|n}^{\alpha}$ of $\mathfrak{gl}_{m|n}$ with generators $E_{i,j}(d)$ ($1 \leq i,j \leq m+n$ and $d \in \mathbb{Z}$) subject to the following relation:
\begin{multline*}
	[E_{i,j}(d), E_{p,q}(d')] \\
	= \delta_{pj}E_{i,q}(d+d')-\delta_{iq}(-1)^{(\pari{i}+\pari{j})(\pari{p}+\pari{q})}E_{p,j}(d+d') + d \delta_{d+d',0}\Big( \delta_{iq}\delta_{jp}(-1)^{\pari{i}}\alpha + \delta_{ij}\delta_{pq}(-1)^{\pari{i}+\pari{p}} \Big).
\end{multline*}
An algebra automorphism $\eta_{\beta}$ ($\beta \in \mathbb{C}$) of $U(\hat{\mathfrak{gl}}_{m|n}^{\alpha})$ is defined by
\[
	E_{i,j}(d) \mapsto E_{i,j}(d) + \delta_{d,0}\delta_{i,j}(-1)^{\pari{i}}\beta.
\]

We define a $\mathbb{Z}/2\mathbb{Z}$-grading of $\frak{g} = \mathfrak{gl}_{M|N} \cong \mathfrak{gl}_{l} \otimes \mathfrak{gl}_{m|n}$ by
\begin{equation}
	\frak{g}_{\pari{j}} = \mathfrak{gl}_{l} \otimes \left( \mathfrak{gl}_{m|n} \right)_{\pari{j}} \label{eq:parity}
\end{equation}
for $j=0,1$.
Then the parity of $E_{s,t} \otimes E_{i,j}$ is $\pari{i}+\pari{j}$.
This is not the standard parity if $l \geq 2$.
One can define the rectangular $W$-superalgebra $\mathcal{W}^{\kappa}(\mathfrak{g},f)$ attached with
\begin{equation*}
	\frak{g} = \mathfrak{gl}_{M|N} \cong \mathfrak{gl}_{l} \otimes \mathfrak{gl}_{m|n}, \ f = \sum_{s=1}^{l-1} \sum_{i=1}^{m+n} E_{s(m+n)+i,(s-1)(m+n)+i} = \left( \sum_{s=1}^{l-1} E_{s+1,s} \right) \otimes I_{m+n},
\end{equation*}
and a supersymmetric bilinear form $\kappa$ on $\frak{g} = \mathfrak{sl}_{M|N} \oplus \mathfrak{z}_{M+N}$ defined by
\begin{equation*}
	\kappa(X,Y) = \begin{cases}
		k \str_{\frak{g}}(XY) & \text{if } X,Y \in \mathfrak{sl}_{M \mid N},\\
		(k+M-N)(M-N) & \text{if } X=Y = I_{M+N},\\
		0 & \text{if } X \in \mathfrak{sl}_{M \mid N} \text{ and } Y \in \mathfrak{z}_{M+N}.
	\end{cases}
\end{equation*}
Note that the supertrace is taken with respect to the parity given by (\ref{eq:parity}).
The explicit value of $\kappa$ is
\[
	\kappa(E_{s_1,t_1} \otimes E_{i,j}, E_{s_2,t_2} \otimes E_{p,q}) = \delta_{s_1,t_2}\delta_{t_1,s_2} \delta_{iq}\delta_{jp}(-1)^{\pari{i}}k + \delta_{s_1,t_1}\delta_{s_2,t_2} \delta_{ij}\delta_{pq}(-1)^{\pari{i}+\pari{p}}.
\]
See \cite{MR4088277} and \cite{Affine_super_Yangians_and_rectangular_W-superalgebras} for details.
Its current algebra $\mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f))$ is identified with a subalgebra of $U(\hat{\mathfrak{gl}}_{m|n}^{\alpha})_{\mathrm{comp}}^{\otimes l}$ via the Miura map.
The injectivity of the Miura map for $W$-superalgebras has been established by Genra \cite{MR3663604} for generic levels and by Nakatsuka \cite{2005.10472} for general levels.

By \cite{Affine_super_Yangians_and_rectangular_W-superalgebras}, one can define generators $W_{i,j}^{(r)}(d)$ ($1 \leq r \leq l, \ 1 \leq i,j \leq m+n, \ d \in \mathbb{Z}$) of $\mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f))$ as follows.
Let $A^{[s]} = \left( (-1)^{\pari{i}}E_{i,j}^{[s]}[-1]_{(-1)} \right)_{1 \leq i,j \leq m+n}$ be the $(m+n) \times (m+n)$ matrix whose $(i,j)$-entry is $(-1)^{\pari{i}}E_{i,j}^{[s]}[-1]_{(-1)}$.
We define a matrix $W^{(r)}$ by
\begin{equation*}
	(\tau + A^{[1]}) (\tau + A^{[2]}) \cdots (\tau + A^{[l]}) = \tau^{l} + \sum_{r=1}^{l} \tau^{l-r} W^{(r)}.
\end{equation*}
Then generators $W_{i,j}^{(r)}$ of the rectangular $W$-superalgebra $\mathcal{W}^{\kappa}(\mathfrak{g},f)$ are defined by
\[
	W^{(r)} = \left( (-1)^{\pari{i}}W_{i,j}^{(r)} \right)_{1 \leq i,j \leq m+n}.
\]
We set
\[
	W_{i,j}^{(r)}(d) = W_{i,j}^{(r)} t^{d+r-1} \in \mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f)).
\]
Moreover, if $\alpha \neq 0$ then the elements of the generators for $r=1,2$ are enough to generate $\mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f))$.
\begin{exam}\label{exam:generators_super}
We have
\begin{align*}
	W_{i,j}^{(1)}(d) &= \sum_{s=1}^{l} E_{i,j}^{[s]}(d),\\
	W_{i,j}^{(2)}(d) &= \sum_{r \in \mathbb{Z}} \sum_{1 \leq s_1 < s_2 \leq l} \sum_{a=1}^{m+n} (-1)^{\pari{a}} E_{i,a}^{[s_1]}(d+r) E_{a,j}^{[s_2]}(-r) - (d+1)\alpha \sum_{s=1}^{l} (l-s) E_{i,j}^{[s]}(d).
\end{align*}
\end{exam}

We similarly introduce parabolic inductions $\Delta_{l_1,l_2}$ and $\tilde{\Delta}_{l_1,l_2}$ for a decomposition $l=l_1+l_2$.
They are injective homomorphisms from $\mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f))$ to $\mathfrak{U}(\mathcal{W}^{\kappa_1}(\mathfrak{gl}_{M_1 \mid N_1}, f_1)) \otimes_{\mathrm{comp}} \mathfrak{U}(\mathcal{W}^{\kappa_2}(\mathfrak{gl}_{M_2 \mid N_2}, f_2))$, where $M_i = l_i m$, $N_i = l_i n$ ($i=1,2$).
An algebra automorphism $\eta^{(l)}$ of $U(\hat{\mathfrak{gl}}_{m|n}^{\alpha})_{{\mathrm{comp}}}^{\otimes l}$ is defined by the same way as in the non-super case.

The affine super Yangian $Y(\hat{\frak{sl}}_{m|n})$ was defined by the second named author in \cite{Affine_super_Yangian}.
See also \cite[C.2]{2011.01203}.
It has two parameters $\hbar, \varepsilon$ as the affine Yangian $Y(\hat{\frak{sl}}_{n})$.
Its generators are denoted by $X_{i,r}^{+}$, $X_{i,r}^{-}$, $H_{i,r}$ ($i = 0,1,\ldots,m+n-1$ and $r \in \mathbb{Z}_{\geq 0})$.
Set
\begin{gather*}
	X_{0}^{+} = E_{m+n,1}(1), \quad X_{0}^{-} = -E_{1,m+n}(-1), \quad H_{0} = (-E_{m+n,m+n}) - E_{1,1} + c,\\
	X_{i}^{+} = E_{i,i+1}, \quad X_{i}^{-} = (-1)^{\pari{i}} E_{i+1,i}, \quad H_{i} = (-1)^{\pari{i}}E_{i,i}-(-1)^{\pari{i+1}}E_{i+1,i+1} \ \ (i \neq 0)
\end{gather*}
in the affine Lie superalgebra $\hat{\mathfrak{sl}}_{m|n}$ and use the same symbols for their images in $Y(\hat{\frak{sl}}_{m|n})$.

We prepare notations in order to simplify formulas below.
For $i=1,\ldots,m+n-1$ and $a=1,\ldots,m+n$, we set
\[
	\cpari{i}{a} = \pari{a} + (\pari{i}+\pari{a})(\pari{i+1}+\pari{a}) = \begin{cases}
		0 & \text{ if } 1 \leq i \leq m-1, \\
		\pari{a} & \text{ if } i=m, \\
		1 & \text{ if } m+1 \leq i \leq m+n-1.
	\end{cases}
\]
Fix a sequence $(c_{i})_{1 \leq i \leq m+n-1}$ by $c_{i}=-i/2$ for $1 \leq i \leq m$ and $c_{m+i}=(-m+i)/2$ for $1 \leq i \leq n-1$.
We give formulas for the coproduct $\Delta \colon Y(\hat{\mathfrak{sl}}_{m|n}) \to Y(\hat{\mathfrak{sl}}_{m|n})_{\mathrm{comp}}^{\otimes 2}$ and the evaluation map $\ev \colon Y(\hat{\mathfrak{sl}}_{m | n}) \to U(\hat{\mathfrak{gl}}_{m|n}^{\alpha})_{\mathrm{comp}}$ from \cite{Affine_super_Yangian}.
The latter map exists under the assumption $\alpha + m - n = -\varepsilon/\hbar$.
We omit to write down their explicit forms for $H_{i,1}$ here;
\begin{align*}
	&\Delta(X_{0,1}^{+}) = \square(X_{0,1}^{+}) \\
	&\ + \hbar \Bigg( X_{0}^{+} \otimes c + \sum_{d \geq 0} \sum_{a=1}^{m+n} (-1)^{\pari{a}} \Big( E_{a,1}(d+1) \otimes E_{m+n,a}(-d) - E_{m+n,a}(d+1) \otimes E_{a,1}(-d) \Big)\Bigg),\\
	&\Delta(X_{0,1}^{-}) = \square(X_{0,1}^{-}) \\
	&\ + \hbar \Bigg( c \otimes X_{0}^{-} - \sum_{d \geq 0} \sum_{a=1}^{m+n} (-1)^{\pari{a}} \Big( E_{a,m+n}(d) \otimes E_{1,a}(-d-1) - E_{1,a}(d) \otimes E_{a,m+n}(-d-1) \Big)\Bigg)
\end{align*}
and
\begin{align*}
	&\Delta(X_{i,1}^{+}) = \square(X_{i,1}^{+}) + \hbar \sum_{d \geq 0} \\
	&\quad\Bigg( \sum_{a=1}^{i} (-1)^{\cpari{i}{a}} E_{a,i+1}(d) \otimes E_{i,a}(-d) + \sum_{a=i+1}^{m+n} (-1)^{\cpari{i}{a}} E_{a,i+1}(d+1) \otimes E_{i,a}(-d-1) \\
	&\qquad\qquad\quad - \sum_{a=1}^{i} (-1)^{\pari{a}} E_{i,a}(d+1) \otimes E_{a,i+1}(-d-1) - \sum_{a=i+1}^{m+n}  (-1)^{\pari{a}} E_{i,a}(d) \otimes E_{a,i+1}(-d) \Bigg),\\
	&\Delta(X_{i,1}^{-}) = \square(X_{i,1}^{-}) + (-1)^{\pari{i}}\hbar \sum_{d \geq 0} \\
	&\quad\Bigg( \sum_{a=1}^{i} (-1)^{\cpari{i}{a}} E_{a,i}(d) \otimes E_{i+1,a}(-d) + \sum_{a=i+1}^{m+n} (-1)^{\cpari{i}{a}} E_{a,i}(d+1) \otimes E_{i+1,a}(-d-1) \\
	&\qquad\qquad\quad - \sum_{a=1}^{i} (-1)^{\pari{a}} E_{i+1,a}(d+1) \otimes E_{a,i}(-d-1) - \sum_{a=i+1}^{m+n} (-1)^{\pari{a}} E_{i+1,a}(d) \otimes E_{a,i}(-d) \Bigg)
\end{align*}
for $i=1,\ldots,m+n-1$;
\begin{align*}
		\ev(X_{0,1}^{+}) &= \hbar \Bigg( \alpha X_{0}^{+} + \sum_{d \geq 0} \sum_{a=1}^{m+n} (-1)^{\pari{a}} E_{m+n,a}(-d) E_{a,1}(d+1) \Bigg),\\
		\ev(X_{0,1}^{-}) &= \hbar \Bigg( \alpha X_{0}^{-} - \sum_{d \geq 0} \sum_{a=1}^{m+n} (-1)^{\pari{a}} E_{1,a}(-d-1) E_{a,m+n}(d) \Bigg)
\end{align*}
and
\begin{align*}
		&\ev(X_{i,1}^{+}) = \hbar \times\\
		&\ \Bigg( c_{i} X_{i}^{+} + \sum_{d \geq 0} \Bigg( \sum_{a=1}^{i} (-1)^{\pari{a}} E_{i,a}(-d) E_{a,i+1}(d) + \sum_{a=i+1}^{m+n} (-1)^{\pari{a}} E_{i,a}(-d-1) E_{a,i+1}(d+1) \Bigg) \Bigg),\\
		&\ev(X_{i,1}^{-}) = \hbar \times\\
		&\ \Bigg( c_{i} X_{i}^{-} + (-1)^{\pari{i}} \sum_{d \geq 0} \Bigg( \sum_{a=1}^{i} (-1)^{\pari{a}} E_{i+1,a}(-d) E_{a,i}(d) + \sum_{a=i+1}^{m+n} (-1)^{\pari{a}} E_{i+1,a}(-d-1) E_{a,i}(d+1) \Bigg) \Bigg)
\end{align*}
for $i=1,\ldots,m+n-1$.

\begin{dfn}\label{dfn:map_super}
Assume $k+M-N=\alpha+m-n=-\varepsilon/\hbar$.
We set
\[
	\Phi_{l} = \eta^{(l)} \circ \ev^{\otimes l} \circ \Delta^{l-1},
\]
which is an algebra homomorphism from $Y(\hat{\frak{sl}}_{m|n})_{\mathrm{comp}}$ to $U(\hat{\mathfrak{gl}}_{m|n}^{\alpha})_{{\mathrm{comp}}}^{\otimes l}$.
\end{dfn}

In the computation of $\Phi_{l}$, we use the formula
\[
	\square(X) \square(Y) = \square(XY) + X \otimes Y + (-1)^{|X||Y|} Y \otimes X,
\]
where $|X|$ and $|Y|$ denote their parities.
We omit details.
\begin{prop}\label{prop:image_super}
We have
\begin{equation*}
	\begin{split}
		&\Phi_{l}(X_{0,1}^{+}) = (-\hbar) \Bigg( W_{m+n,1}^{(2)}(1) - \alpha W_{m+n,1}^{(1)}(1) - \sum_{d \geq 0} \sum_{a=1}^{m+n} (-1)^{\pari{a}} W_{m+n,a}^{(1)}(-d) W_{a,1}^{(1)}(d+1)\Bigg),\\
		&(-1)^{\pari{m+n}} \Phi_{l}(X_{0,1}^{-}) \\
		&= (-\hbar) \Bigg( W_{1,m+n}^{(2)}(-1) - l \alpha W_{1,m+n}^{(1)}(-1) - \sum_{d \geq 0} \sum_{a=1}^{m+n} (-1)^{\pari{a}} W_{1,a}^{(1)}(-d-1) W_{a,m+n}^{(1)}(d) \Bigg)
	\end{split}
\end{equation*}
and
\begin{equation*}
	\begin{split}
		&\Phi_{l}(X_{i,1}^{+}) = (-\hbar) \Bigg( W_{i,i+1}^{(2)}(0) - c_{i} W_{i,i+1}^{(1)}(0) \\
		&\qquad\quad - \sum_{d \geq 0} \Bigg( \sum_{a=1}^{i} (-1)^{\pari{a}} W_{i,a}^{(1)}(-d) W_{a,i+1}^{(1)}(d) + \sum_{a=i+1}^{m+n} (-1)^{\pari{a}} W_{i,a}^{(1)}(-d-1) W_{a,i+1}^{(1)}(d+1) \Bigg)\Bigg),\\
		&(-1)^{\pari{i}} \Phi_{l}(X_{i,1}^{-}) = (-\hbar) \Bigg( W_{i+1,i}^{(2)}(0) - c_{i} W_{i+1,i}^{(1)}(0) \\
		&\qquad\quad - \sum_{d \geq 0} \Bigg( \sum_{a=1}^{i} (-1)^{\pari{a}} W_{i+1,a}^{(1)}(-d) W_{a,i}^{(1)}(d) + \sum_{a=i+1}^{m+n} (-1)^{\pari{a}} W_{i+1,a}^{(1)}(-d-1) W_{a,i}^{(1)}(d+1) \Bigg)\Bigg)
	\end{split}
\end{equation*}
for $i=1,\ldots,m+n-1$.
\end{prop}
Then we obtain a result in the super case by the same argument as in the non-super case.
\begin{thm}\label{thm:introduction_main_super}
Assume $k+M-N=\alpha+m-n=-\varepsilon/\hbar$.
Then the map $\Phi_{l}$ gives an algebra homomorphism
\[
	\Phi_{l} \colon Y(\hat{\frak{sl}}_{m|n})_{\mathrm{comp}} \to \mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f))
\]
which makes the following diagram commutative:
\[
	\xymatrix{
		Y(\hat{\frak{sl}}_{m|n}) \ar[d]_{\Delta} \ar[r]^{\Phi_{l}} \ & \ \mathfrak{U}(\mathcal{W}^{\kappa}(\mathfrak{g},f)) \ar[d]^{\tilde{\Delta}_{l_1,l_2}} \\
		Y(\hat{\frak{sl}}_{m|n})^{\otimes 2}_{\mathrm{comp}} \ar[r]^-{\Phi_{l_1} \otimes \Phi_{l_2}} \ \ & \ \ \mathfrak{U}(\mathcal{W}^{\kappa_1}(\mathfrak{gl}_{M_1 \mid N_1}, f_1)) \otimes_{\mathrm{comp}} \mathfrak{U}(\mathcal{W}^{\kappa_2}(\mathfrak{gl}_{M_2 \mid N_2}, f_2)).
	}
\]
Moreover, $\Phi_{l}$ is surjective if $\alpha = k+(M-N)-(m-n) \neq 0$.
\end{thm}

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text:
Meinke is a surname. Notable people with the surname include:

Bob Meinke (1887–1952), American baseball player
Frank Meinke (1863–1931), American baseball player
Katrin Meinke (born 1979), German cyclist
Nina Meinke (born 1993), German boxer
Peter Meinke (born 1932), American author

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Huge Police Chase in Dallas/Garland?
Discussion in 'Texas' started by Jared, Oct 9, 2009.
Any news links? I wanna see this.
was it on 30 this morning?
:lmao: I want to see a video.
i watched it live. it was pretty funny.
I've chased 2 F-150s that body style at LOW speeds!! Its something about those trucks that make you wanna run!!
what a moron... said he didnt want to go to jail thats why he ran.
I think it would be cool to see if i could get away. Ya know, like in and abandoned city or something.
Disclaimer: I would NEVER run from the police or endanger innocent people.


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Pelham Middle School is an International Baccalaureate World School offering the IB Middle Years Programme to all of its students. The International Baccalaureate (or "IB") works with schools to help develop well-rounded students who respond to challenges with optimism and an open mind, are confident in their own identities, make ethical decisions, join with others in celebrating our common humanity, and are prepared to apply what they learn in real-world, complex, and unpredictable situations.
The Middle Years Programme (or "MYP") provides a framework of learning that encourages students to become creative, critical, and reflective thinkers. By emphasizing intellectual challenge and encouraging students to make connections between their studies in traditional subjects and the real world, the MYP fosters the development of skills for communication, intercultural understanding, and global engagement: essential qualities for young people who will one day become global leaders.
For the past three years, our teachers have participated in extensive professional development through which they learned how to plan curriculum using the Middle Years Programme framework, worked on aligning the curriculum across grade levels and subjects, explored strategies in the areas of formative assessment and differentiated instruction, planned inquiry-based lessons and authentic, project-based assessments, and created rubrics to more accurately assess student progress and communicate with students and parents about learning outcomes.
During the coming year, teachers will continue planning units that engage students as inquirers and thinkers and foster a stimulating learning environment based on understanding and respect. We will also continue to expand our use of the MYP Assessment Criteria to assess student work in order to provide a feedback process that is more standardized, precise, transparent, and holistic: one that not only reports on student learning but encourages it. In addition, we will strive to further incorporate the Approaches to Learning Skills (helping students "learn how to learn") and Service as Action (through which the curriculum inspires students to take principled action in their local and global communities).
For more information about IB, visit www.ibo.org. If you have any additional questions, please contact Sean A. Llewellyn, Interim Assistant Principal and IB Coordinator, at sllewellyn@pelhamschools.org, or call (914) 738-8190.


text:
[ G.R. No. 183233, December 23, 2009 ]
VIRGILIO G. ANABE, PETITIONER, VS. ASIAN CONSTRUCTION (ASIAKONSTRUKT), ZENAIDA P. ANGELES AND N.O. GARCIA, RESPONDENTS.
CARPIO MORALES, J.:
Virgilio G. Anabe (petitioner) was hired by respondent Asian Construction (Asiakonstrukt) as radio technician/operator on April 15, 1993. By notice dated September 8, 1999, he was advised that his services would be, as he was in fact, terminated effective October 8, 1999 on the ground of retrenchment. Petitioner thus filed on February 10, 2000 a complaint[1] for illegal dismissal and illegal deduction and payment of overtime pay, premium pay, holiday pay, service incentive leave pay, and 13th month pay.
Asiakonstrukt, attributing petitioner's retrenchment to sudden business reversal in the construction industry, averred, however, that petitioner's money claims have been offset against his outstanding accountabilities.
By Decision[2] of June 29, 2001, the Labor Arbiter, finding that Asiakonstrukt failed to submit financial statements to prove losses, ruled that petitioner was not validly dismissed. Thus he disposed:
WHEREFORE, premises considered, judgment is hereby rendered finding the respondents liable for illegal dismissal and consequently ordered to reinstate complainant to his former position or its equivalent without loss of seniority rights and other privileges, with full backwages and benefits from date of dismissal up to actual date of reinstatement which is in the amount of P136,277.14 as of this month. Respondent[s] are likewise ordered to pay complainant his 13th month pay in the amount of P4,259.64 and illegal deductions in the amount of P164,960.24 and overtime pay in the amount of P6.11 [underpayment of overtime pay as computed by the Computation and Examination Unit of the NLRC]. Respondents are further ordered to pay complainant ten percent (10%) of the total award as attorney's fees.
On appeal, the National Labor Relations Commission (NLRC), taking into consideration the certified true copies of the Audited Financial Statements from 1998 to 2000 submitted by Asiakonstrukt, partly granted the appeal by Resolution[3] of March 10, 2004. It modified the Labor Arbiter's Decision by holding that petitioner was not illegally dismissed. While it affirmed the award of the 13th month pay, overtime pay and attorney's fees, it ordered the payment to petitioner of P19,170 as separation pay.
Moreover, the NLRC reduced the reimbursable amount of illegal deductions from P164,960.24 to P88,000.00, ratiocinating that petitioner is only entitled to money claims from 1997-1999, the claims prior thereto having already prescribed.
Petitioner's motion for reconsideration was denied by Order[4] dated August 31, 2005, hence, he appealed to the Court of Appeals, assailing the consideration by the NLRC of the Audited Financial Statements which were submitted only on appeal.
By Decision[5] of December 26, 2007, the appellate court held that there was no grave abuse of discretion on the part of the NLRC when it considered the financial statements as they "already form part of the records on appeal."
Citing Clarion Printing House, Inc. v. NLRC,[6] the appellate court noted that the NLRC is not precluded from receiving evidence on appeal as technical rules of procedure are not binding in labor cases. And it affirmed the ruling of the NLRC that petitioner is only entitled to the illegal deductions for the period 1997-1999 in the amount of P88,000.00, as the prescriptive period for money claims is only three years from the time the cause of action accrues.
Petitioner's motion for reconsideration having been denied by Resolution[7] of April 2, 2008, he filed the present petition, maintaining that he was illegally dismissed as Asiakonstrukt failed to prove that it was suffering business losses to warrant a valid retrenchment of its employees; and Asiakonstrukt belatedly submitted financial statements were not shown to be newly found evidence and unavailable during the proceedings before the Labor Arbiter to thus cast doubts as to their veracity.
The petition is partly meritorious.
Retrenchment is the termination of employment initiated by the employer through no fault of and without prejudice to the employees, it is resorted to during periods of business recession, industrial depression, or seasonal fluctuations or during lulls occasioned by lack of orders, shortage of materials, conversion of the plant for a new production program or the introduction of new methods or more efficient machinery or of automation. It is a management prerogative resorted to, to avoid or minimize business losses,[8] and is recognized by Article 283 of the Labor Code, as amended, viz:
Art. 283. Closure of establishment and reduction of personnel.--The employer may also terminate the employment of any employee due to x x x retrenchment to prevent losses or the closing or cessation of operations of the establishment x x x by serving a written notice on the worker and the [DOLE] at least one month before the intended date thereof. x x x In case of retrenchment to prevent losses, the separation pay shall be equivalent to one (1) month pay or at least one-half month pay for every year of service whichever is higher. x x x (Emphasis ours.)
To effect a valid retrenchment, the following elements must be present: (1) the retrenchment is reasonably necessary and likely to prevent business losses which, if already incurred, are not merely de minimis, but substantial, serious, and real, or only if expected, are reasonably imminent as perceived objectively and in good faith by the employer; (2) the employer serves written notice both to the employee/s concerned and the Department of Labor and Employment at least a month before the intended date of retrenchment; (3) the employer pays the retrenched employee separation pay in an amount prescribed by the Code; (4) the employer exercises its prerogative to retrench in good faith; and (5) the employer uses fair and reasonable criteria in ascertaining who would be retrenched or retained.[9]
The losses must be supported by sufficient and convincing evidence,[10] the normal method of discharging which is the submission of financial statements duly audited by independent external auditors.[11]
In the present case, Asiakonstrukt failed to submit its audited financial statements within the two years that the case was pending before the Labor Arbiter. It submitted them only after it received the adverse judgment of the Labor Arbiter.
Indubitably, the NLRC is not precluded from receiving evidence on appeal as technical rules of evidence are not binding in labor cases. There is, however, a caveat to this policy. The delay in the submission of evidence should be clearly explained and should adequately prove the employer's allegation of the cause for termination.[12] In the present case, Asiakonstrukt proffered no explanation behind the belated submission. And the financial statements[13] it submitted covered the period 1998-2000. Further, note that the audited financial statement[14] covering the period 1998-2000 was prepared in April 2001, which begs the question of how the management knew at such date of the company's huge losses to justify petitioner's retrenchment in 1999.
Furthermore, from the certification[15] issued by the Securities and Exchange Commission (SEC), it would appear that Asiakonstrukt failed to submit its financial statements to the SEC, as required under the law, for the period 1998-2000 and 2003-2005, thereby lending credence to petitioner's theory that the financial statements submitted on appeal may have been fabricated. Indeed, Asiakonstrukt could have easily submitted its audited financial statements during the pendency of the proceedings at the labor arbiter's level, especially considering that it was in late 2001 that the case was decided.
For failure then of Asiakonstrukt to clearly and satisfactorily substantiate its financial losses,[16] the dismissal of petitioner on account of retrenchment is unjustified. Petitioner is thus entitled to the twin reliefs of payment of backwages and other benefits from the time of his dismissal up to the finality of this Court's Decision, and reinstatement without loss of seniority rights or, in lieu thereof, payment of separation pay.
On the reduction of petitioner's money claims on account of prescription, under Article 1139 of the Civil Code, actions prescribe by the mere lapse of the time prescribed by law. That law may either be the Civil Code or special laws as specifically mandated by Article 1148. In labor cases, the special law on prescription is Article 291 of the Labor Code which provides:
Article 291. Money Claims. - All money claims arising from employer-employee relations accruing during the effectivity of this Code shall be filed within three (3) years from the time the cause of action accrued; otherwise they shall be barred forever. (emphasis supplied)
The Labor Code has no specific provision on when a monetary claim accrues. Thus, again the general law on prescription applies. Article 1150 of the Civil Code provides that -
Article 1150. The time for prescription for all kinds of actions, when there is no special provision which ordains otherwise, shall be counted from the day they may be brought. (emphasis supplied)
The day the action may be brought is the day a claim started as a legal possibility.[17] In the present case, the day came when petitioner learned of Asiakonstrukt's deduction from his salary of the amount of advances he had received but had, by his claim, been settled, the same having been reflected in his payslips, hence, it is assumed that he learned of it at the time he received his monthly paychecks.
As thus correctly ruled by both the NLRC and the appellate court, only those illegal deductions made from 1997 to 1999 when he was dismissed can be claimed, he having filed his complaint only in February 2000. Per his own computation and as properly adopted by the NLRC in its assailed Resolution dated March 10, 2004, petitioner is thus entitled to reimbursement of P88,000.00.
WHEREFORE, the petition is GRANTED. The Court of Appeals Decision dated December 26, 2007 and Resolution dated April 2, 2008 are SET ASIDE. The Decision of the Labor Arbiter dated June 29, 2001 is REINSTATED, with the MODIFICATION that petitioner, Virgilio G. Anabe, is entitled to P88,000.00 representing reimbursement of the illegal deductions from his salary.
The case is REMANDED to the National Labor Relations Commission which is DIRECTED to recompute WITH DISPATCH the monetary awards due petitioner.
Puno, C.J., (Chairperson), Velasco, Jr.*, Leonardo-De Castro, and Bersamin, JJ., concur.
* Additional Member per Raffle dated December 14, 2009.
[1] NLRC records I, p. 1.
[2] Rollo, pp. 85-99. Penned by Labor Arbiter Aliman Mangandog.
[3] CA rollo, pp. 19-24. Penned by Commissioner Romeo L. Go and concurred in by Presiding Commissioner Roy V. Señeres.
[4] Id. at 26-30. Penned by Commissioner Romeo L. Go and concurred in by Presiding Commissioner Benedicto Ernesto R. Bitonio, Jr. and Commissioner Perlita B. Velasco.
[5] Id. at 829-838. Penned by Associate Justice Sesinando E. Villon and concurred in by Associate Justices Martin S. Villarama, Jr. (now Associate Justice of this Court) and Noel G. Tijam.
[6] G.R. No. 148372, June 27, 2005, 461 SCRA 272.
[7] CA rollo, p. 1034. Penned by Associate Justice Sesinando E. Villon and concurred in by Associate Justices Martin S. Villarama, Jr. (now Associate Justice of this Court) and Noel G. Tijam.
[8] Mobilia Products, Inc. v. Demecilio, G.R. No. 170669, February 4, 2009.
[9] Vide Asian Alcohol Corporation v. NLRC, G.R. No. 131108, March 25, 1999, 305 SCRA 416, 428.
[10] Guerrero v. National Labor Relations Commission, G.R. No. 119842, August 30, 1996, 261 SCRA 301, 305.
[11] Vide F.F. Marine Corporation v. National Labor Relations Commission, Second Division, G.R. No. 152039, April 8, 2005, 455 SCRA 154, 168.
[12] Vide AG & P United Rank & File Association v. NLRC, 332 Phil. 937 (1996).
[13] NLRC records II, pp. 78-94.
[15] Vide certification dated September 8, 2005, CA rollo, pp. 849-850; certification dated January 14, 2008, CA rollo, p. 1028.
[17] Paras, CIVIL CODE OF THE PHILIPPINES, 14th Ed., Vol. IV, p. 60.

redpajama_set_name: RedPajamaCommonCrawl


text:
Mystery plaintiff sues school district over cellphone records
STORY BY RAY MCNULTY
An anonymous plaintiff has filed a lawsuit alleging the school district has withheld public records – specifically text messages, emails, photos, notes and logs – from board member Tiffany Justice's district-issued cellphone.
The lawsuit, filed last week, is seeking a court order to compel the district to provide to the plaintiff the documents, including any communications that have been deleted from her iPhone as well as those stored in the iCloud account that backs up the device.
In particular, the plaintiff's lawsuit targets phone and message logs for Justice's communications with Sheriff's Maj. Eric Flowers and former Vero Beach Mayor Val Zudans between December 2018 and October 2019.
Flowers, who is a candidate for sheriff, launched a four-month cyber-stalking investigation requested by Justice earlier this year, when a school district employee posted tweets implying Justice had an inappropriate relationship with now-former Schools Superintendent Mark Rendell.
Justice and Flowers are friends and she has publicly endorsed him in the sheriff's race. They are known to have exchanged numerous text messages about the cyberstalking investigation.
There is no obvious similar connection between Justice and Zudans.
"I don't really have anything to do with the School Board, so I don't know what they're looking for," Zudans said, adding he and Justice are friendly but not close friends. "It's probably a fishing expedition by somebody who doesn't like the three of us."
The plaintiff, identified only as "John Doe, aka saveverobeach@gmail.com" in the court filing, claims the district has refused or ignored numerous and repeated requests for the documents made last month under Florida's public records law.
In addition to a court order for the requested records, the plaintiff also is asking the judge to order the district to pay his attorney's fees, administrative costs and any "further relief as this court may deem appropriate," the lawsuit states.
The school district, school district superintendent (now Dr. David Moore), School Board and Justice are named as defendants in the lawsuit.
School district Public Information Officer Cristen Maddux said the district "does not comment on open litigation." Justice did not respond to a text message seeking comment.
The plaintiff's Rockledge-based attorney, Catherine Jones, did not return multiple phone calls to her law office.
The case has been assigned to Circuit Court Judge Janet Croom in Indian River County.
The lawsuit states the plaintiff made "numerous public records requests" on Nov. 12 and Nov. 13 related to Justice's use of her district-provided cellphone, electronic mail account and social media accounts, but the district had produced only one of the many records sought at the time the lawsuit was filed.
According to the lawsuit, Justice received the record requests and forwarded them to Brenda Davis, the superintendent's administrative assistant and designated "custodian" of public records requests presented to the district.
As of last week, the records still hadn't been provided.
Under Chapter 119 of the Florida Statutes – more commonly known as the "Public Records Law" – any records made or received by any public agency in the course of its official business are available for inspection, unless specifically exempted by the Legislature.
Those records include tapes, voice recordings, photographs, film and records stored in computers, cellphones and other devices.