text:
\section{Introduction}

Let $ M$ be an $m$ dimensional manifold and let $\omega^i$ be a collection of  $n$ differential forms
on $M$ satisfying
\begin{equation}
d\omega^i +\frac{1}{2} C^i_{jk} \omega^j \wedge \omega^k =0
\EqTag{FSE}
\end{equation}
where $C^i_{jk}$ are the structure constants of an $n$-dimensional real Lie algebra $\lieg$.  It is well known (see \cite{sharpe:1997a} or \cite{griffiths:1974a}) that given $x_0\in M$ there exists an open set $U$ about $x_0$ and a smooth map $\rho:M \to G$ where $G$ is a Lie group with Lie algebra $\lieg$ such that
\begin{equation}
\omega^i = \rho^* \tau^i
\EqTag{MF1}
\end{equation}
where $\tau^i$ are a basis of left invariant forms on $G$.  If $M$ is simply connected then $U$ can be taken to be all of $M$. These facts and many others related to equations \EqRef{FSE} and \EqRef{MF1} are proved in Chapter 3 of \cite{sharpe:1997a}.

Equation \EqRef{FSE} occurs in a number of applications such as the integration of Pfaffian systems of finite type with symmetry, see \cite{lychagin:1991a}, \cite{doubrov:2000a} and \cite{fels:2007a} and the reconstruction problem for quotients of Pfaffian systems \cite{anderson-fels:2005a}, \cite{fels:2007a}.   Another example where equation \EqRef{FSE} occurs  is Lie's third theorem. One version of this theorem (see Section \StRef{SLieT}) states that given a Lie algebra $\lieg$ with structure constants $C^i_{jk}$, there exists $n$ point-wise linearly independent one-forms $\omega^i$ on $\real^n$ satisfying equation \EqRef{FSE}.

\smallskip

Our interest lies in how quadrature and matrix algebra can be used to solve the problems listed above when $\lieg$ is a solvable Lie algebra. By quadrature we mean the following (see \cite{lychagin:1991a}). Given a closed differential one-form $\omega$ on a simply connected manifold $M$, then $M  = df $ for a smooth function $f:M \to \real$ and $f$ is determined from $\omega$ by quadrature.

Central to our approach is a simple reduction theorem proved in Section \StRef{Sreduce}  (Theorem \StRef{reduce}). Theorem \StRef{reduce} shows that for a Lie algebras $\lieg$ which admit a codimension one ideal $\liek \subset \lieg$, the structure equations \EqRef{FSE} can be reduced to those of the co-dimension one ideal in $\lieg$ using one quadrature and the matrix exponential. A simple induction argument then applies to solvable Lie algebras which reduces the structure equations \EqRef{FSE} to $d \check \omega^i = 0$, where $\check \omega^i$ are invertible matrix combinations of $\omega^i$ which are found using $n$ quadratures (Corollary \StRef{C2}).

In Section \StRef{Sfi} we show how these results can be used to construct first integrals for a completely integrable Pfaffian system with a solvable symmetry algebra using quadrature and matrix exponentiation.

In Section \StRef{SLieT} we reverse the argument from Corollary \StRef{C2} in Section \StRef{Sreduce} and this gives an algebraic proof of Lie's Third theorem for vector-fields. More specifically we give a closed form formula for the $n$ vector-fields on $\real^n$ in terms of matrix exponentials (Theorem \StRef{PLT} and Corollary \StRef{CLV}). In comparison the proof given by Cartan of this version of Lie's third theorem which is valid for any Lie algebra uses ODE existence theorems (see  \cite{flanders:1989a}) and does not give closed form formulas.

In Section \StRef{Lie3G} our main goal in this section is to show how the multiplication map $\mu:G \times G\to G$ for a simply connected solvable Lie group $G$ can be found by $n$-quadratures and the matrix exponential (Theorem \StRef{L3M}). In order to do this we first return to the problem of finding the map $\rho:M \to G$  satisfying \EqRef{MF1} when $G$ is solvable and $M$ is simply connected. The result here turns out to be surprisingly simple. Corollary \StRef{C2} of Section \StRef{Sreduce} and Theorem \StRef{PLT} in Section \StRef{SLieT} automatically produce the map $\rho:M \to G$ (Theorem \StRef{CMF}). In particular the algorithm from Section \StRef{Sreduce} that reduces the structure equations \EqRef{FSE} for $\omega^i$ to  $d \check \omega^i=0$, produces the map $\rho:M \to G$ (Theorem \StRef{CMF}).  Therefore Theorem \StRef{CMF} produces the map $\rho:M \to G$ for solvable $G$ using only quadratures and the matrix exponential.

Our  final result in Section \StRef{Lie3G} takes a simply connected any simply connected solvable Lie group $G$ and uses Lie's third theorem from Section \StRef{SLieT} to  construct on $G\times G$ a set of $n$ one forms $\omega^i$ satisfying equations \EqRef{FSE} with the property that the map $\rho:G\times G \to G$ in Theorem \StRef{CMF} is in fact the multiplication map (Theorem \StRef{L3M}). This gives an algorithm that produces the global multiplication map $\mu:G \times G \to G$ for any $n$-dimensional simply connected solvable Lie group using at most $n$ quadratures and products of matrix exponentials.

A  number of examples which demonstrates the theory are given throughout.

\section{Reduction Theorem}\StTag{Sreduce}

Let $\lieg$ be an $n$-dimensional Lie algebra which admits a codimension one ideal $\liek\subset \lieg$. Theorem \StRef{reduce} given below allows in this case a reduction of the differential forms satisfying equation \EqRef{FSE} to the structure constants of the ideal $\liek$ using one quadrature (the homotopy operator for closed one-forms) and the matrix exponential. This theorem is hinted at in \cite{lychagin:1991a} and \cite{doubrov:2000a} but is not mentioned explicitly or proved.

Suppose that $\liek \subset \lieg$ is a codimension 1 ideal in the $n$-dimensional Lie algebra $\lieg$. A basis $\beta=\{ \bfe_1,\ldots , \bfe_{n-1},\bfe_n\}$  for $\lieg$ such that $\hat \beta = \{ \bfe_a \}_{1\leq  a\leq n-1}$ is a basis for $\liek$ is {\bf adapted to the codimension one ideal $\liek\subset \lieg$}. In this case the structure constants are
\begin{equation}
[\bfe_a, \bfe_b ]= C_{ab}^c\bfe_c, \quad [\bfe_n, \bfe_a] = C_{na}^b \bfe_b\quad ,   1\leq a,b,c \leq n.
\EqTag{SEab}
\end{equation}
Note that $C^a_{bc}$ in equation \EqRef{SEab} are the structure constants of the $n-1$ dimensional Lie algebra $\liek$ in the basis $\hat \beta$.

This leads to a well-known lemma whose proof is a simple exercise,  see  \cite{lychagin:1991a}.

\begin{Lemma}  \StTag{WK} Let $\liek \subset \lieg$ be a codimension one ideal and let $\beta=\{ \bfe_a,\bfe_n\}_{1\leq a \leq n}$
be adapted basis.  Let  $\tilde \omega^i$ be $n$ differential forms  on a manifold $M$ satisfying equation \EqRef{FSE} for the Lie algebra $\lieg$ where the structure constants $\tilde C^i_{jk}$ are given in a basis $\tilde \beta=\{ {\bf f}_i \}_{1\leq i \leq n}$, and let $[P^i_j]$ be the change of basis matrix
$$
{\bf f}_j=P^i_j {\bf e}_i.
$$
Then $ \omega^i=P^i_j \tilde \omega^j$ satisfy the structure equations
\begin{equation}
\begin{aligned}
 d\omega^a +\frac{1}{2} C^a_{bc} \omega^b \wedge \omega^c + C^a_{nb} \omega^n \wedge \omega^b & = 0\\
 d \omega^n &  = 0
\end{aligned}
\EqTag{ASE}
\end{equation}
where $C^a_{bc}$ and $C^n_{nb}$ are the structure constants in the basis $\beta$ (equation \EqRef{SEab}).
\end{Lemma}

\medskip
Note that Lemma \StRef{WK} that forms $\omega^i$ satisfying equation \EqRef{FSE}
for a solvable Lie algebra $\lieg$ can be chosen to satisfy equations \EqRef{ASE} by a constant linear change in the forms $\omega^i$. This leads to the following fundamental reduction theorem which is central to this article is the following.

\begin{Theorem} \StTag{reduce} Suppose $\lieg$ is an $n$-dimensional Lie algebra with $\liek \subset \lieg$ a codimension one ideal and
let $\beta=\{\bfe_a, \bfe_n\}_{1\leq a \leq n-1}$ be basis adapted to the codimension one ideal $\liek \subset \lieg$.   Let  $\bfomega=[ \omega^1,\ldots, \omega^n]^T$ be  an $n$-vector of differential one-forms  on a simply connected manifold $M$ satisfying \EqRef{ASE} where $C^i_{jk}$ are the structure constants of $\lieg$ in the basis $\beta$.

Let $f:M \to \real$ so that $\omega^n $ in \EqRef{ASE} satisfies
\begin{equation}
\omega^n= df,
\EqTag{int}
\end{equation}
and  let $\bfhomega$ be the vector of $n$ one-forms on $M$ defined by
\begin{equation}
\bfhomega = \rme^{f [\text{\rm ad} ({\bf e}_n)]}\cdot \bfomega,
\EqTag{homega}
\end{equation}
where $[\text{\rm ad} ({\bf e}_n)] $ is the $n$ by $n$ matrix representation of $\text{\rm ad} ({\bf e}_n)$ in the basis $\beta$. Then the structure equations for $\bfhomega$ are
\begin{equation}
d \homega^a + \frac{1}{2} C^a_{bc} \homega^b \wedge \homega^c =0\, , \quad d \homega^n  = 0
\EqTag{FSE3}
\end{equation}
where $C^a_{bc}$ are the structure constants for $\liek$ in the basis $\hat \beta = \{ {\bf e}_a\}_{1\leq a\leq n-1}$.
\end{Theorem}

\begin{proof}   The matrix representation of the derivation $\text{\rm ad}({\bf e}_n)$  is given in the adapted basis from equation \EqRef{SEab} by computing,
\begin{equation}
\text{\rm ad}( {\bf e}_n) (\be_b)= [\be_n, \be_b] =C_{nb}^a \be_a, \ \text{\rm ad}( {\bf e}_n) (\be_n) =[\be_n,\be_n]= 0.
\EqTag{aden}
\end{equation}
Let $\widetilde \text{\rm ad}(\be_n) : \liek \to \liek $ be the restriction of $\text{\rm ad}({\bf e}_n)$ to the invariant subspace $\liek$. The matrix representation $[\, \widetilde \text{\rm ad}(\be_n)\, ]$in the basis $\beta$ is given using equation \EqRef{aden} by
\begin{equation}
[\widetilde \text{\rm ad}(\be_n)]^a_b= C^a_{nb}.
\EqTag{alpc}
\end{equation}
The matrix valued function $ \rme^{ f\, [\,\text{\rm ad}({\bf e}_n)\, ]} $ in equation \EqRef{homega} can then be written,
\begin{equation}
\rme^{ f\,[\, \text{\rm ad}({\bf e}_n)\, ]}  =  \left[ \begin{array}{cc} \rme^{ f\, [\, \widetilde \text{\rm ad}(\be_n)\, ]  }  & {\bf 0} \\ {\bf 0}^T & 1 \end{array} \right]
\EqTag{Ainbas}
\end{equation}
where ${\bf 0}$ is the $n-1$ zero vector and $T$ is the transpose. The forms $\bfhomega$ in equation \EqRef{homega} can then be written using equation \EqRef{Ainbas} (and matrix vector multiplication) as
\begin{equation}
[\homega^a] = \rme^{ f \,[\, \widetilde \text{\rm ad}(\be_n)\,] } [\omega^b]  , \quad \homega^n = \omega^n .
\EqTag{homega2}
\end{equation}

The verification of equation \EqRef{FSE3} in Theorem \StRef{reduce} can now be checked directly by taking the exterior derivative of the forms $\homega^a$ in equation \EqRef{homega2}. In order to do so we need the exterior derivative of $\rme^{ f\, [\, \widetilde \text{\rm ad}(\be_n)\, ]}$  from equation \EqRef{alpc} is
\begin{equation}
\begin{aligned}
d \ \rme^{ f\, [\, \widetilde \text{\rm ad}(\be_n)\, ]}   = df \,  \rme^{ f\, [\, \widetilde \text{\rm ad}(\be_n)\, ]} [\, \widetilde \text{\rm ad}(\be_n)\, ]
\EqTag{dA}
\end{aligned}
\end{equation}
We now compute $d \homega^a$ using equations \EqRef{dA}, \EqRef{ASE}, \EqRef{alpc} and \EqRef{int},
\begin{equation}
\begin{aligned}
& &[ d  \homega^a ]  &=   \rme^{ f\, [\, \widetilde \text{\rm ad}(\be_n)\, ]} [\, \widetilde \text{\rm ad}(\be_n)\, ][-\omega^b\wedge df]+ \rme^{ f\, [\, \widetilde \text{\rm ad}(\be_n)\, ]}[ d \omega^b]\\
& & & =   \rme^{ f\, [\, \widetilde \text{\rm ad}(\be_n)\, ]}[ - C^c_{nb} \omega^b \wedge df]  - \rme^{ f\, [\, \widetilde \text{\rm ad}(\be_n)\, ]}[ \frac{1}{2} C^b_{cd} \omega^c \wedge \omega^d + C^b_{cn}\omega ^c \wedge  \omega^n]\\
& & &=   \rme^{ f\, [\, \widetilde \text{\rm ad}(\be_n)\, ]}[- C^c_{nb}  \omega^b\wedge \omega^n] -  \rme^{ f\, [\, \widetilde \text{\rm ad}(\be_n)\, ]}
[ \frac{1}{2} C^b_{cd} \omega^c \wedge \omega^d + C^b_{cn}\omega ^c \wedge  \omega^n] \\
& & & =  \rme^{ f\, [\, \widetilde \text{\rm ad}(\be_n)\, ]}[-
 \frac{1}{2} C^b_{cd} \omega^c \wedge \omega^d ]\\
& && =[-\frac{1}{2} C^a_{cd}  \homega^c \wedge \homega^d],
\end{aligned}
\end{equation}
where the last line follows because $\rme^{ f\, [\, \widetilde \text{\rm ad}(\be_n)\, ]} $ is the matrix representation of an automorphism of $\liek$ for each $x\in M$.
\end{proof}

Theorem \StRef{reduce} involves one quadrature in equation \EqRef{int} and then only the matrix exponential \EqRef{homega}. We also note that the change of forms in equation \EqRef{homega} results in the elimination of the third term in the first structure equation in \EqRef{ASE}.  Since the constants  $C^a_{bc}$ in equation \EqRef{FSE3} are the structure constants for $\liek$ in the basis $\{ \bfe_a \}_{1\leq  a\leq n-1}$, equation \EqRef{homega} can then be thought of reducing the Lie algebra $\lieg$ to $\liek$.  See also Remark \StRef{BFR} below.

We now apply Theorem \StRef{reduce} to the case where $\lieg$ admits a sequence of subalgebras $\liek_s\subset \lieg$, $s=0,\ldots,r$
\begin{equation}
\liek_r \subset \liek_{r-1} \subset \ldots
\liek_1 \subset \liek_0= \lieg
\EqTag{sid}
\end{equation}
where each $\liek_s \subset \liek_{s-1}$, $1\leq s \leq r$ is a codimension 1 ideal. The ideals $ \liek_s,  s=0,\ldots, r$ in equation \EqRef{sid} are called {\bf a sequence of codimension one ideals}. Note that $\dim \liek_s = \dim \lieg -s =n-s$. A basis $\beta= \{ \bfe_i \}_{1\leq i \leq n} $ for $\lieg$ is said to be {\bf adapted to a sequence of codimension one ideals} $\liek_s \subset \lieg, s=0,\ldots,r$ if
\begin{equation}
\liek_s =\spn \{\bfe_1,\ldots, \bfe_{n-s} \}, \quad s=0\ldots r.
\EqTag{Abas}
\end{equation}
Given the basis $\beta= \{ \bfe_i \}_{1\leq i \leq n} $ adapted to the sequence of codimension one ideal $\liek_s\subset \lieg$ we  let $\text{\rm ad}_{s}( \bfe_{n-s}) \in {\rm der}(\liek_s)$ be the restriction,
\begin{equation}
\text{\rm ad}_{s}( \bfe_{n-s}) = \text{\rm ad}( \bfe_{n-s}) |_{\liek_s}, \quad 0\leq s\leq r-1.
\EqTag{alseq}
\end{equation}
In particular $\text{\rm ad}_0(\bfe_n) = \text{\rm ad}(\bfe_n)$.

Theorem \StRef{reduce} easily extends to the case of a sequence of codimension one ideals to the following.

\begin{Corollary} \StTag{C1} Suppose the Lie algebra $\lieg$ in Theorem \StRef{reduce} admits a sequence of codimension one ideals $\liek_s \subset \lieg,\ s=0,\ldots,r $,and let $\beta= \{ \bfe_i \}_{1\leq i \leq n}$ be a basis adapted to the sequence.
Then there exists $r$ functions $ f^t:M \to \real, \ t= n-r+1,\ldots, \leq n $ and $n-r$ differential one-forms $\{ \comega^\alpha \}_{1\leq \alpha \leq n-r}$ obtained inductively from $\omega^i$ by $r$ quadratures and the exponential of matrices such that

\begin{equation}
\left[ \begin{array}{cc}  \rme^{f^{n-r+1}[\text{\rm ad}_{r-1}( \bfe_{n-r+1})]} & {\bf 0}_{r-1} \\ {\bf 0}_{r-1}^T & {\rm I}_{r-1} \end{array} \right]\cdots\left[ \begin{array}{cc}  \rme^{f^{n-1}[\text{\rm ad}_{1}( \bfe_{n-1})]} & {\bf 0}_1 \\ {\bf 0}_1 ^T & 1 \end{array} \right] \, \rme^ {f^n[\text{\rm ad}(\be_n)]} \,
 \left[ \begin{array}{c} \omega^1\\ \vdots \\ \omega^n  \end{array} \right]
= \left[ \begin{array}{c} \comega^1\\ \vdots \\ \comega^{n-r} \\ d  f ^{n-r+1} \\ \vdots \\ df^n  \end{array} \right]
\EqTag{seqreduce}
\end{equation}
where $[\text{\rm ad}_{s}( \bfe_{n-s})]$ is the $n-s$ by $n-s$ matrix representation of $\text{\rm ad}_{s}( \bfe_{n-s}) \in {\der}(\liek_s)$ defined in \EqRef{alseq} in the basis \EqRef{Abas} for $\liek_{s}$, ${\rm I}_s $ is the $s$ by $s $ identity matrix and ${\bf 0}_s$ is the $n-s$ by $s$ zero matrix.

Furthermore the $n-r$ differential one forms $\comega^\alpha$ satisfy
\begin{equation}
d \comega^\alpha +\frac{1}{2}C^\alpha_{\beta \gamma} \comega^\beta \wedge \comega^\gamma = 0
\EqTag{redste}
\end{equation}
where $C^\alpha_{\beta \gamma}$ are the structure constants for the $n-r$-dimensional Lie algebra $\liek_r$ in the basis $\check \beta=\{ \bfe_\alpha\}_{1\leq \alpha \leq n-r}$
\end{Corollary}

\begin{proof} Using Theorem \StRef{reduce} we inductively compute the functions $f^t$, $n-r+1\leq t \leq n$. Let $f^n:M \to \real$ be the function obtained in Theorem \StRef{reduce} using $\liek=\liek_1$ so that $ \omega^n=df^n$. The first term is
\begin{equation}
 \rme^{ f^n [\, \text{\rm ad}({\bf e}_n)\,]  } \left[ \begin{array}{c} \omega^1\\ \vdots \\ \omega^n \end{array} \right].
 = \left[ \begin{array}{c} \homega^1\\ \vdots \\ \homega^{n-1} \\ df^n  \end{array} \right].
\EqTag{CBn0}
\end{equation}

Now because $\liek_{2} \subset \liek_{1}$ is a codimension one ideal, and the basis $\{\bfe_i\}_{1\leq i\leq n}$ is adapted to the sequence \EqRef{sid}, we have
$$
d \homega^{n-1} = 0.
$$
Again letting $f^{n-1} :M \to \real$ so that $\homega^{n-1} = d f^{n-1}$, and applying Theorem \StRef{reduce} to the case of $\liek_2 \subset \liek_1$ we get
\begin{equation}
\rme^{ f^{n-1} [\, \text{\rm ad}_1(\be_{n-1})\, ] }
\left[ \begin{array}{c} \homega^1 \\  \vdots \\ \homega^{n-1} \end{array} \right]=
 \left[ \begin{array}{c} \tomega^1 \\ \vdots \\ \tomega^{n-2} \\ df^{n-1}  \end{array} \right]
\EqTag{CBn1}
\end{equation}
where the $n-2$ differential forms $\tomega^u$ satisfy
$$
d \tilde \omega^u +\frac{1}{2}   C^u_{vw} \tilde \omega^u \wedge \tilde \omega^v = 0
$$
where $C^u_{vw}$ are the structure constants for $\liek_2$ in the basis $\tilde \beta=\{\be_u\}_{1\leq u \leq n-2}$.
Using equation \EqRef{CBn1} and equation \EqRef{CBn0} produces the right most two matrices on the left hand side of equation \EqRef{seqreduce}.

Continuing by induction this produces, by $n-r$ quadratures, the $n-r$ functions $f^t:M \to \real$ and the matrices in equation \EqRef{seqreduce}. Furthermore the forms $\bfcomega$ in equation \EqRef{seqreduce} satisfy the structure equation \EqRef{redste}. \end{proof}

It is well known that if $\lieg$ is an $n$ dimensional solvable Lie algebra, then $\lieg$ admits a sequence of codimension 1 ideals $\liek_s, \ s= 0,\ldots,n$, \cite{jacobson:1962a}, and these can be found using elementary linear algebra (for example by taking a basis adapted to the derived series). This leads to the next corollary.

\begin{Corollary} \StTag{C2} Let $\lieg$ be an $n$ dimensional solvable  Lie algebra $\lieg$ and let $\beta=\{ \bfe_i\}_{1\leq i \leq n}$ be a basis adapted to a sequence of codimension one ideals $\liek_s$, $s=0,\ldots,n$. There exists $n$ functions $f^i:M \to \real$, $i=1,\ldots,n$,   obtained inductively from the forms $\omega^i$ by $n$ quadratures and the exponential of matrices so that
\begin{equation}
\left[ \begin{array}{cc}  \rme^{f^{2}[\, \text{\rm ad}_{n-2}(\be_2)\,]} & {\bf 0}_{n-2} \\ {\bf 0}^T_{n-2} & {\rm I}_{n-2} \end{array} \right] \cdots
\left[ \begin{array}{cc}  \rme^{f^{n-1}[\text{\rm ad}_{1}( \bfe_{n-1})]} & {\bf 0}_{1} \\ {\bf 0}_{1} ^T & 1 \end{array} \right] \, \rme^ {f^n[\text{\rm ad}(\be_n)]} \,
 \left[ \begin{array}{c} \omega^1\\ \vdots \\ \omega^n  \end{array} \right]
  = \left[ \begin{array}{c}  d  f^1\\ \vdots \\df^n \end{array} \right],
\EqTag{FIE}
\end{equation}
where $[\text{\rm ad}_{s}( \bfe_{n-s})]$ is the $n-s$ by $n-s$ matrix representation of $\text{\rm ad}_{s}( \bfe_{n-s}) \in {\der}(\liek_s)$ defined in \EqRef{alseq} in the basis \EqRef{Abas} for $\liek_{s}$ where $ s=0,\ldots,n-2$, ${\rm I}_s $ is the $s$ by $s$ identity matrix, and ${\bf 0}_s$ is the $n-s$ by $s$ zero matrix.
\end{Corollary}

\begin{Remark} The function $f:M \to \real$ in Theorem \StRef{reduce} is uniquely determined if the value of $f$ is prescribed at a point. Similarly for the functions in Corollaries \StRef{C1} and \StRef{C2}.
\end{Remark}

\begin{Remark} \StTag{BFR} An alternative description of Theorem \StRef{reduce} is the following. Given differential one-forms $\{ \omega^i \}_{ 1\leq i \leq n}$ on a manifold $M$ satisfying \EqRef{FSE} in terms of the basis $\beta=\{ \be_i\}_{1\leq i \leq n}$ for the Lie algebra $\lieg$,  define the $\lieg$ valued differential one-form $\bfomega_{\lieg}$ by
$$
\bfomega_\lieg = \omega^i \otimes \be_i .
$$
Equation \EqRef{FSE} is then written
\begin{equation}
d \bfomega_\lieg + \frac{1}{2} \bfomega_\lieg \wedge \bfomega_\lieg =0
\EqTag{FSE2}
\end{equation}
where the wedge product $ \bfomega_\lieg \wedge_\lieg \bfomega_\lieg $ is defined by
\begin{equation}
\bfomega_\lieg\wedge \bfomega_\lieg (X,Y) = [\bfomega_\lieg(X), \bfomega_\lieg(Y)]_{\lieg} \qquad {\rm for \ all \ }  X,Y \in TM.
\EqTag{LBE}
\end{equation}

Suppose now that $\lieg$ is a Lie algebra and $\liek\subset \lieg$ is a codimension one ideal and that $\bfomega_\lieg$ is a Lie algebra valued form on a simply connected manifold $M$ satisfying the structure equations $d\bfomega_\lieg+\frac{1}{2} \bfomega_\lieg\wedge \bfomega _\lieg=0$.
Let $A :M \to {\rm Aut}(\lieg)$ be $ A(x) = \exp\left(f(x) \text{\rm ad}({\bf e}_n)\right)$ where $f:M \to \real$ is
defined in \EqRef{int}.  Theorem \StRef{reduce} shows
 that  $\hbfomega_{\liek \times \real }= A\circ \omega $ takes values in the Lie algebra $\liek \times \real$ and satisfies
$$
d \hbfomega_{\liek \times \real} + \frac{1}{2} \hbfomega_{\liek \times \real} \wedge \hbfomega_{\liek \times \real} =0.
$$
The induction in Corollaries \StRef{C1} and \StRef{C2} can then by applied to $\hbfomega_{\liek \times \real }$.
\end{Remark}

\section{First integrals for completely integrable Pfaffian systems with a solvable symmetry algebra} \StTag{Sfi}

Let $I \subset T^*M$ be a rank $r$ Pfaffian system. A function $f:M\to \real$ such that $df(x)\in I_x$ for all $x\in M$ is called a {\bf first integral} of $I$, see \cite{lychagin:1991a} and \cite{doubrov:2000a}. Suppose  $I= \spn \{\omega^i\}$ and $\omega^i$ satisfy the
structure equations \EqRef{FSE} then Theorem \StRef{reduce} and its corollaries can be used to compute first integrals using only quadrature and matrix exponentiation.  Theorem \StRef{reduce} states that given a co-dimension one ideal in $\lieg$, one quadrature produces the first integral $f$ in equation \EqRef{int} for the completely integrable Pfaffian system $I=\spn \{ \omega^i \}$ and the exponential of a matrix reduces the structure equations \EqRef{FSE} to the structure equations of the ideal $\liek$.  Corollary \StRef{C2} gives rise to the following.

\begin{Corollary} \StTag{S2C1} Let $I\subset T^*M$ be a completely integrable Pfaffian system where  $I= \spn \{\omega^i\}$ and
$$
d\omega^i + \frac{1}{2} C^i_{jk} \omega^j \wedge \omega^k = 0,
$$
where $C^i_{jk}$ are the structure constants of a solvable Lie algebra $\lieg$. Then there exists $n$ first integrals $f^i : M \to \real$ of $I$ which can be constructed sequentially by quadratures and matrix exponentials through formula \EqRef{FIE}.
\end{Corollary}

We now describe how Corollary \StRef{S2C1} is used in practice see \cite{lychagin:1991a}. Let $\Gamma=\{X_i\}_{1\leq i \leq n} $ be a Lie algebra of vector fields on $M$ which are infinitesmimal symmetries of a completely integrable rank $n$ Pfaffian system $I=\spn \{ \theta^j\}_{1\leq j \leq n} $. Assume
 $\bfGamma_p=\spn\{X_i(p)\}$ is a $n$-dimensional subspace of $T_pM$ for each $p\in M$ which satisfies the transversality condition
\begin{equation}
 \bfGamma_p \cap \ann(I_p) = 0, \qquad {\rm for \ all } \ p \in M.
\EqTag{tc}
\end{equation}
The transversality condition \EqRef{tc} implies that $n\times n$ matrix $(P^i_j)= (\theta^i(X_j))$ is invertible, and a simple
computation (see \cite{lychagin:1991a} or \cite{fels:2007a}) shows that the differential forms
\begin{equation}
\omega^i = (P^{-1})^i_j \theta^j
\EqTag{domega}
\end{equation}
satisfy $I = \spn\{ \omega^i \}$ and
$$
d \omega^i +\frac{1}{2} C^i_{jk} \omega^j \wedge \omega^k  = 0
$$
where $[X_j,X_k]= C^i_{jk} X_i $.  If the Lie algebra $\Gamma$ is solvable then Corollary \StRef{C2} may be used to produce $n$ first integrals for $I$.

\begin{Example}\StTag{Ex1}
Consider the ODE  from pg. 152 in \cite{olver:1998a},
\begin{equation}
u_{xxx}=\frac{3u_{xx}^2}{u_x}+\frac{u_{xx}^3}{u_x^5} \EqTag{621}
\end{equation}
On the  manifold $ {M} = \{ (x,u,u_x,u_{xx}) \in \real^4 \ | \ u_x \neq 0 \
\} $, equation \EqRef{621} defines the Pfaffian system $I=\spn\{ \theta^1,\theta^2,\theta^3\}$ where
\begin{equation}
 \theta^1= du -u_x dx,\quad  \theta^2=du_x -u_{xx}dx,\quad \theta^3=
 du_{xxx} - \left( 3u_{xx}^2u_x^{-1}+u_{xx}^3u_x^{-5}\right) dx.
 \EqTag{PS3}
\end{equation}
The Pfaffian system \EqRef{PS3} admits the infinitesimal symmetries,
\begin{equation}
Z_1  = \partial_x, \ Z_2 = \partial_u ,\ Z_3 = u\partial_x
-u_x^2\partial_{u_x} -3u_xu_{xx}\partial_{u_{xx}} \
\EqTag{SVF}
\end{equation}
and the only non-vanishing bracket is
\begin{equation}
[Z_2, Z_3]= Z_1.
\EqTag{MCE}
\end{equation}
Let $\{ \, \be_1,\, \be_2,\, \be_3\, \}$ be a basis for the three dimensional Lie algebra isomorphic to the
Lie algebra of vector-fields $\{ Z_1,Z_2,Z_3 \}$ so that $[\be_2,\be_3]=\be_1$.  We have the sequence of codimension one ideals $\spn\{\be_1\} \subset \spn\{\be_1, \be_2\} \subset \lieg$.

On the set ${ M}^0=(x,u,u_x\neq 0,u_{xx}\neq 0)$
the forms  $ \{\omega^i\}_{1\leq i\leq 3} $ on ${ M}^0$ are computed from equation \EqRef{domega} using equations \EqRef{PS3} and  \EqRef{SVF} to be
\begin{equation}
\begin{split} \omega^1 &=
dx+\frac{1}{(u_xu_{xx})^2}(3u_x^6+3u_{xx}uu_x^4+u_{xx}^2u)du_x-\frac{u_x^3}{u_{xx}^3}(u_x^2+u_{xx}u)du_{xx}
, \\ \omega^2 & = du+\frac{3
u_x^5}{u_{xx}^2}du_x-\frac{u_x^6}{u_{xx}^3}du_{xx}  ,\quad  \omega^3 =
-\frac{(3u_x^4+u_{xx})}{u_{xx}u_x^2}du_x+\frac{u_x^3}{u_{xx}^2}du_{xx}\
. \end{split}
\EqTag{omex1}
\end{equation}

Now $ d\omega^3 =0$ and we may write $\omega^3 = df^3$ where
\begin{equation}
f^3 =   \frac{1}{u_x}-\frac{u_x^3}{u_{xx}} .
\EqTag{om3}
\end{equation}
in equation \EqRef{FIE}. Then by exponentiating
\begin{equation}
[\, \text{\rm ad}(\be_3)\,] = \left[ \begin{array}{ccc} 0 & -1 & 0  \\ 0 & 0 & 0 \\ 0 &  0  & 0  \end{array} \right]
\EqTag{ad3}
\end{equation}
this gives us our first step in the algorithm of Corollary \StRef{C2} as
\begin{equation}
\left[ \begin{array}{ccc} 1 & -f^3 & 0 \\ 0 & 1 & 0 \\ 0 & 0 &  1  \end{array} \right] \left[ \begin{array}{c} \omega^1 \\ \omega^2 \\ \omega^3 \end{array} \right]
= \left[ \begin{array}{c} \hat \omega^1 \\ \hat \omega^2\\ df^3 \end{array} \right]
\end{equation}
where
$$
\begin{aligned}
\left[ \begin{array}{cc} \hat \omega^1 \\ \hat \omega^2 \end{array} \right] & = \left[ \begin{array}{cc} 1 &   \frac{u_x^3}{u_{xx}}- \frac{1}{u_x} \\ 0 & 1 \end{array} \right]
\left[ \begin{array}{cc}  \omega^1 \\  \omega^2 \end{array} \right] \\
& = \left[ \begin{array}{cc}  dx+\frac{u_x^4-u_{xx}}{u_x u_{xx}} du+\frac{3 u_x^{10}+3 u u_{xx}^2 u_x^4+u u_{xx}^3}{u_{xx}^3 u_x^2} du_x -\frac{u_x^3(u_x^6+u u_{xx}^2)}{u_{xx}^4} d u_{xx}\\  du+\frac{3 u_x^5}{u_{xx}^2} du_x-\frac{u_x^6}{u_{xx}^3} du_{xx} \end{array} \right].
\end{aligned}
$$
Now
$$
d \hat \omega^1 =0, \quad d \hat \omega^2 = 0.
$$
The corresponding first integrals $f^1, f^2: M^0 \to \real$ satisfying the equations $ \hat \omega^1 =df^1, \hat \omega^2 = df^2$ are given by
\begin{equation}
f^1 = \frac{u_x^{10}+3u u_{xx}^2u_x^4+3 x u_x u_{xx}^3-3 u u_{xx}^3}{ 3 u_ x u_{xx}^3}\ , \quad
f^2 = \frac{u_x^6+2 u u_{xx}^2}{2 u_{xx}^2}.
\EqTag{ex1f1f2}
\end{equation}
\end{Example}

\section{Lie's Third Theorem: The Vector-field Case}\StTag{SLieT}

The global vector-field version of Lie's third theorem is the following.

\begin{Theorem} (Lie's third theorem for vector-fields) Let $\lieg$ be an $n$-dimensional Lie algebra. There exists $n$ pointwise linearly independent vector-fields on $\real^n$ which form a Lie algebra of vector-fields that is isomorphic to $\lieg$.
\end{Theorem}

Starting with a solvable Lie algebra $\lieg$ we give a proof of this version of Lie's third theorem by constructing the vector-fields using only the matrix exponential. The proof follows almost directly from Corollary \StRef{C2}. First we give the dual version.

\begin{Theorem} \StTag{PLT} Let $\lieg$ be an $n$-dimensional solvable Lie algebra and  let  $\beta= \{ \bfe_i \}_{1\leq i \leq n} $ be a basis for $\lieg$  adapted to a sequence of codimension one ideals $\liek_s\subset \lieg$, $s=0,\ldots,n$. Let $\text{\rm ad}_s(\be_{n-s}) \in {\der}(\liek_s), s=0,\ldots, n-2$ be the derivation defined in \EqRef{alseq} by
\begin{equation}
\text{\rm ad}_s(\be_{n-s}) =  \text{\rm ad} ( \bfe_{n-s})|_{\liek_{s}} , \quad s=0,\ldots, n-2.
\EqTag{alseq2}
\end{equation}
Let $\tau^i$ be the  $n$ point-wise linear independent one-forms  on $\real^n$ defined by
\begin{equation}
\left[ \begin{array}{c} \tau^1 \\ \vdots \\ \tau^n \end{array} \right] =
 \rme^{-x^{n}[\text{\rm ad}(\be_n) ]}
   \left[ \begin{array}{cc} \rme^{-x^{n-1}[\text{\rm ad}_{1}{(\be_{n-1})}]} & {\bf 0}_1 \\ {\bf 0}_1^T & 1 \end{array} \right]
 \ldots
\left[ \begin{array}{cc} \rme^{-x_{2}[\text{\rm ad}_{n-2}(\be_2)]} & {\bf 0}_{n-1} \\ {\bf  0}_{n-1}^T & {\rm I}_{n-1} \end{array} \right]
\left[ \begin{array}{c} dx^1 \\ \vdots \\ dx^n \end{array} \right]
\EqTag{Lieom}
\end{equation}
where $[\text{\rm ad}_s(\be_{n-s})]$ is the $n-s$ by $n-s$ matrix representation of $\text{\rm ad}_s(\be_{n-s})$ in the basis  \EqRef{Abas} for $\liek_{s}$, $s=0,\ldots,n-2$,  ${\rm I}_{s}$ is the $s$ by $s$ identity matrix, and ${\bf 0}_s$ is the $n-s$ by $s$ zero matrix.
Then $\tau^i$ satisfy
\begin{equation}
d \tau^i +\frac{1}{2}C^i_{jk} \tau^j \wedge \tau^k = 0,
\EqTag{LTF}
\end{equation}
where $C^i_{jk}$ are the structure constants for $\lieg$ in the basis $\beta$.
\end{Theorem}

Before presenting the proof of Theorem \StRef{PLT}, let $X_i, 1\leq i \leq n$ be the dual frame to $\tau^i$ in equation \EqRef{Lieom}. This produces the $n$ linearly independent vector-fields on $\real^n$, given by
\begin{equation}
\left[ \begin{array}{c} X_1 \\ \vdots \\ X_n \end{array} \right] =
{\rm e}^{x_n[\text{\rm ad}(\be_n)]^T}\left[ \begin{array}{cc} \rme^{x^{n-1}[\text{\rm ad}_{1}{(\be_{n-1})}]^T} & {\bf 0}_1 \\ {\bf 0}_1^T & 1 \end{array} \right] \cdots \left[ \begin{array}{cc} {\rm e}^{x_{2}[\text{\rm ad}_{n-2}(\be _2)]^T } & {\bf 0}_{n-1} \\ {\bf 0}^T_{n-1} & {\rm I}_{n-1}   \end{array}\right]    \left[ \begin{array}{c} \partial_{x^1} \\ \vdots \\ \partial_{x^n} \end{array} \right],
\EqTag{LieX}
\end{equation}
where $T$ is the transpose. We then have the following, which is Lie's third theorem for vector-fields.

\begin{Corollary}\StTag{CLV}  The  $n$ vector-fields on $X_i$ on $\real^n$ defined in \EqRef{LieX} are point-wise linearly independent and satisfy
\begin{equation}
[X_i,X_j]=C^k_{ij} X_k
\EqTag{BLA}
\end{equation}
where $C^i_{jk}$ are the structure constants for $\lieg$ in the adapted basis $\beta$ of Theorem \StRef{PLT}.
Therefore the vector-fields in \EqRef{LieX} produce a solution to Lie's third Theorem (for vector-fields) using only the matrix exponential where the linear map $\phi:\lieg \to \spn\{ X_i\}$ induced by
\begin{equation}
\phi({\bf e}_i) = X_i
\EqTag{giso}
\end{equation}
is a Lie algebra isomorphism.
\end{Corollary}

\begin{proof}(Theorem \StRef{PLT})  The proof of Theorem \StRef{PLT}  is to think of solving equation \EqRef{FIE} of Corollary \StRef{C2} for $\tau^i$. This gives equation \EqRef{Lieom} with $f^i=x^i$.  We give the details since it is handy to have this as a separate construction.

The proof goes by induction and so consider the differential forms on $\real^n$ given by
\begin{equation}
\left[ \begin{array}{c} \htau^1 \\ \vdots \\ \htau^n \end{array} \right] =
 \left[ \begin{array}{cc} \rme^{-x^{n-1}[\text{\rm ad}_{1}{(\be_{n-1})}]} & {\bf 0}_1 \\ {\bf 0}_1^T & 1 \end{array} \right]
 \ldots
\left[ \begin{array}{cc} \rme^{-x_{2}[\text{\rm ad}_{n-2}(\be_2)]} & {\bf 0}_{n-1} \\ {\bf  0}_{n-1}^T & {\rm I}_{n-1} \end{array} \right]
\left[ \begin{array}{c} dx^1 \\ \vdots \\ dx^n \end{array} \right] .
\EqTag{somega}
\end{equation}
By the induction hypothesis, the forms $\{ \htau^a\}_{1\leq a \leq n-1}$ from equation \EqRef{somega} satisfy,
\begin{equation*}
d  \htau^a+ \frac{1}{2} C^{a}_{bc}  \htau^b \wedge  \htau^c=0.
\end{equation*}
where $C^a_{bc}$ are the structure constants of the ideal $\liek_{n-1}$ in the basis $\hat \beta=\{ \be_{a}\}_{1\leq a \leq n-1}$.

The fact that  the differential forms in equation \EqRef{Lieom} satisfy equation \EqRef{LTF} follows simply by reversing the computation in proof of Theorem \StRef{reduce} (in particular using equation \EqRef{dA}). This shows that the forms $\tau^i$ satisfy equation \EqRef{LTF}.  \end{proof}

\begin{Example}\StTag{Ex2} Consider the 5 dimensional solvable Lie algebra $A_{5,ab}$ from \cite{winternitz:1976a}, with multiplication table
\begin{equation}
[\be_1, \be_4] = b\, \be_1,\ [\be_1, \be_5] = a \, \be_1,\ [\be_2, \be_4] = \be_2,\ [\be_2, \be_5] = -\be_3,\ [\be_3, \be_4] = \be_3,\ [\be_3, \be_5] = \be_2,
\EqTag{SCe1}
\end{equation}
where the two real parameters $a,b$ satisfy $a^2+b^2\neq 0$. The basis $\{ \be_i\}_{1\leq i \leq 5}$ is adapted to the sequence of co-dimension one ideals $\liek_s=\spn\{ \be_1,\ldots, \be_{5-s}\}$, $s=0,\ldots,5$.

We find the non-zero $\text{\rm ad}_s(\be_{n-s})$ matrices from equation \EqRef{alseq2} using the structure equations \EqRef{SCe1} to be
\begin{equation}
[\text{\rm ad}(\be_5 )]= \left[ \begin{array}{ccccc} -a & 0 &  0 & 0 &0 \\ 0 & 0 &-1  & 0 & 0 \\ 0 & 1 &0 & 0 & 0 \\  0 & 0  &0 & 0 & 0 \\ 0 & 0  &0 & 0 & 0 \end
{array} \right], \quad
[\text{\rm ad}_1(\be_4)]= \left[ \begin{array}{cccc} -b & 0 &  0 & 0 \\ 0 & -1 & 0  & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0& 0 & 0 \end{array} \right].
\EqTag{Adms}
\end{equation}
The coframe $\bftau$ on $\real^5 $ is computed from \EqRef{Lieom} using equation \EqRef{Adms} and $\text{\rm ad}_3(\be_2)=0$, $\text{\rm ad}_2(\be_3)=0$ is then
\begin{equation}
\begin{aligned}
\left[
\begin{array}{c}
\tau^1 \\ \tau^2 \\ \tau^3 \\ \tau^4 \\ \tau^5 \end{array}\right]
& =
\left[ \begin{array}{ccccc} \rme^{a x^5} & 0 &  0 & 0 & 0 \\ 0 & \cos x^5  & \sin x^5  & 0& 0 \\ 0 & -\sin x^5  & \cos x^5 & 0  & 0\\ 0 & 0  &0 & 1 & 0\\ 0 & 0 & 0& 0 & 1\end{array} \right]
 \left[ \begin{array}{ccccc} \rme^{b x^4} & 0 &  0 &0 & 0  \\ 0 & \rme^{x^4} & 0 &0 & 0  \\ 0 & 0 & \rme^{x^4} &0 & 0   \\ 0 & 0& 0 & 1 & 0 \\ 0 & 0 & 0 & 0& 1   \end{array} \right]
\left[
\begin{array}{c}
dx^1 \\ dx^2 \\ dx^3 \\ dx^4 \\ dx^5 \end{array}\right] \\
&= \left[
\begin{array}{c}
\rme^{a x^5 + bx^4 b} dx^1  \\
\cos x^5\, \rme^{x^4} dx^2+\sin x^5 \, \rme^{x^4} dx^3
\\
 -\sin x^5\, \rme^{x^4} dx^2+\cos x^5 \, \rme^{x^4} dx^3 \\ dx^4 \\ dx^5
\end{array} \right]
\end{aligned}
\EqTag{Ex1forms}
\end{equation}
The structure equations are then easily checked to be,
$$
d\tau^ 1= -b\, \tau^1\wedge \tau^4-a\, \tau^1 \wedge \tau^5,\  d\tau^2 = -\tau^2 \wedge \tau^4-\tau^3 \wedge \tau^5,\ d\tau^3 = \tau^2 \wedge \tau^5-\tau^3 \wedge \tau^4,\ d\tau^4 = 0,\ d\tau^5 = 0.
$$
\end{Example}

\section{Lie's Third Theorem: The Group Case}\StTag{Lie3G}

We begin this section by first showing that $\real^n$ with the vector-fields $\{X_i\}_{1\leq i \leq n}$ in equation \EqRef{LieX} can be given a global Lie group structure. Our ultimate goal is to show how the group multiplication in the next theorem can be obtained by quadratures and matrix exponentiation (Theorem \StRef{L3M}).

\begin{Theorem} \StTag{LG1} Let $\lieg$ be an $n$-dimensional solvable Lie algebra and let $\beta=\{ {\bf e}_i \}_{1 \leq i \leq n}$ be a basis for $\lieg$ adapted to a sequence of codimension one-ideals  $\liek_s\subset \lieg$, $s=0,\ldots,n$. Let $\{X_i\}_{1\leq i \leq n}$ be the vector-fields on $\real^n$ given in equation  \EqRef{LieX}. Given a point $x_0\in \real^n$, there exists a smooth multiplication map $\mu: \real^n \times \real^n\to \real^n$ so that $\real^n$ is a solvable Lie group with identity $x_0$ and with the Lie algebra of left invariant vector-fields $\spn \{ X_i\}_{1\leq i \leq n}$.
\end{Theorem}

Theorem \StRef{LG1} will follow immediately from \cite{palais:1957a} (or page 133 of \cite{bouchetta:2005a}) and the following lemma.

\begin{Lemma}\StTag{CV} The  vector-fields $\{X_i\}_{ 1\leq i \leq n}$ on $\real^n$ in equation \EqRef{LieX} are complete.
\end{Lemma}

\begin{proof} We prove this by induction. Assume that the $n-1$ vector-fields $\{ \tilde X_r\}_{ 1\leq r \leq n-1}$  on $\real^{n-1}$ with coordinates $x^1,\ldots, x^{n-1}$ given by
\begin{equation}
\left[ \begin{array}{c} \tilde X_1 \\ \vdots \\ \tilde X_{n-1} \end{array} \right] = \left[\begin{array}{cc} \rme^{x_{n-1}[\alpha_{n-2}]} & {\bf 0} \\  {\bf 0} & 1 \end{array} \right]^T \ldots \left[ \begin{array}{cc} \rme^{x_{2}[\alpha_1] } & {\bf 0} \\ {\bf 0} & {\rm I}_{n-2}   \end{array}\right]^T  \left[ \begin{array}{c} \partial_{x^1} \\ \vdots \\ \partial_{x^{n-1}} \end{array} \right],
\EqTag{LieX2}
\end{equation}
are complete. The vector-fields $\{ \tilde X_r\}_{1\leq r \leq n-1}$ are globally defined on $\real^{n-1}$ and satisfy equation \EqRef{BLA} with the structure  constants for the co-dimension one sub-algebra $\liek_1 \subset \lieg$ defined in Theorem \StRef{PLT}. By  \cite{palais:1957a} given $\tilde x_0 \in \real^{n-1}$ there exists a Lie group structure on $\real^{n-1}$ with $\tilde x_0$ as the identity such that $\{ \tilde X_r\}_{1\leq r \leq n-1}$ are a basis for the left invariant vector-fields on $\real^{n-1}$.

Now the $n$ vector-fields $X_i$ in \EqRef{LieX}  can be written in terms of the vector-fields in \EqRef{LieX2} considered as vector-fields on $\real^n$ as
\begin{equation}
\left[ \begin{array}{c} X_1 \\ \vdots \\ X_n \end{array} \right] = \rme^{x_n[\text{\rm ad}(\be_n)]^T}  \left[ \begin{array}{c} \tilde X_1 \\ \vdots \\ \tilde X_{n-1} \\ \partial_{x^n} \end{array} \right] .
\EqTag{LieX3}
\end{equation}
Let $\tilde \xi^s_r$ be the coefficients of the vector-fields $\tilde X_ r$  and let $\eta^j_i$ be the coefficients of the vector-fields $X_i$. From equations \EqRef{LieX3} and \EqRef{Ainbas} we have
\begin{equation}
\eta^s_r = \left[ {\rm e}^{x_n[\widetilde \text{\rm ad}(\be_n)]}  \right]^u_r  \tilde \xi^s_u, \quad \eta^s_n =0, \quad \eta^n_r = 0, \quad \eta^n_n= 1.
\EqTag{LieX4}
\end{equation}
The flow equations for $X_r$,  $1\leq r \leq n-1$ on $\real^n$ are determined from \EqRef{LieX4} to be
\begin{equation}
\begin{aligned}
\frac{dx^s}{dt} & = \left[ {\rm e}^{x_n[\widetilde \text{\rm ad}(\be_n]]}  \right]^u_r  \tilde \xi^s _ u\\
\frac{dx^n}{dt} &= 0.
\end{aligned}
\EqTag{Flow1}
\end{equation}
Integrating the last equation in \EqRef{Flow1} gives $x^n= c$. Therefore the first equation in \EqRef{Flow1} can be written
\begin{equation}
\frac{dx^s}{dt}  = K^u_r  \tilde \xi^s _ u
\EqTag{Flow3}
\end{equation}
where $[K^u_r]$ is an invertible matrix.   Equation \EqRef{Flow3} is  the flow equations of the vector-field $\tilde Y_r=K^u_r \tilde X_u$ on $\real^{n-1}$.  For each $r=1,\ldots, n-1$ the vector-field $Y_r$ is a linear combination of the left invariant vector-fields $\tilde X_s$ on $\real^{n-1}$ and is therefore left invariant.  Consequently each $Y_r$ is complete.

Finally we note from equation \EqRef{LieX3}  that $X_n = \partial_{x^n}$ which is complete.\end{proof}

Theorem \StRef{LG1} also proves Lie's third theorem for solvable Lie groups and it gives an alternative proof that a simply connected solvable Lie group is $\real^n$.

We now examine how we can construct the multiplication map $\mu$. In order to
do so, we first return to the original question from the introduction. Let $\{\omega^i\}_{1\leq i \leq n}$ be $n$ differential forms on a simply connected manifold $M$ satisfying equations \EqRef{FIE} (written as $d\bfomega_\lieg + \frac{1}{2} \bfomega_\lieg \wedge \bfomega_\lieg=0$ in Remark \StRef{BFR}). As noted in the introduction, it is well known that there exists a function $ \rho:M \to G $ such that
\begin{equation}
\omega^i= \rho^* \tau^i
\EqTag{MFe}
\end{equation}
where $\tau^i$ are the left invariant Maurer Cartan forms on a Lie group $G$ with Lie algebra $\lieg$ (again see or example \cite{sharpe:1997a}). Furthermore $\rho$ is unique up to left translation by an element of $G$. We now show how Corollary \StRef{C2} combined with Theorem \StRef{LG1} determines $\rho$ in equation \EqRef{MFe} when $\lieg$ is solvable.

\begin{Theorem} \StTag{CMF}   Let $\lieg$ be an $n$-dimensional solvable Lie algebra and let $\beta=\{ {\bf e}_i \}_{1 \leq i \leq n}$ be a basis for $\lieg$ adapted to a sequence of codimension one-ideals  $\liek_s\subset \lieg$, $s=0,\ldots,n$. Let $G=\real^n$ with coordinates $(x^1,\ldots, x^n)$ be the solvable Lie group with basis of left invariant forms $\{ \tau^i \}_{1\leq i \leq n}$ given in \EqRef{Lieom}.

Let $\{\omega^i\}_{1\leq i \leq n}$ be $n$ differential forms on a simply connected manifold $M$ satisfying the structure equations \EqRef{FSE} where $C^i_{jk}$ are computed in the basis $\beta$ for $\lieg$ and let
$\rho : M \to G$ be given by
\begin{equation}
\rho(p) = (x^1= f^1(p), \ldots,x^n= f^n(p) ), \quad {\rm for \ all} \ p \in M
\EqTag{deff1}
\end{equation}
where $f^i:M \to \real$ are the functions from Corollary \StRef{C2}.  Then
\begin{equation}
\rho^* \tau^i = \omega^i.
\EqTag{pbf}
\end{equation}
\end{Theorem}

\begin{proof}  Let $f^i:M \to \real$ be the functions in Corollary \StRef{C2}  so that equation \EqRef{FIE} holds. Solving for $\omega^i$ in equation \EqRef{FIE} gives
\begin{equation}
\left[ \begin{array}{c} \omega^1\\ \vdots \\ \omega^n  \end{array} \right]
=
\rme^ {-f^n[\text{\rm ad}(\be_n)]} \,
\left[ \begin{array}{cc}  \rme^{-f^{n-1}[\text{\rm ad}_{1}( \bfe_{n-1})]} & {\bf 0}_{1} \\ {\bf 0}_{1} ^T & 1 \end{array} \right]\cdots
\left[ \begin{array}{cc}  \rme^{-f^{2}[\, \text{\rm ad}_{n-2}(\be_2)\,]} & {\bf 0}_{n-2} \\ {\bf 0}^T_{n-2} & {\rm I}_{n-2} \end{array} \right]
\left[ \begin{array}{c}  d  f^1\\ \vdots \\df^n \end{array} \right].
\EqTag{MFP1}
\end{equation}
With $\{ \tau^i \}_{1\leq i \leq n}$ whose
formula is in given in  \EqRef{Lieom} in coordinates $(x^i)_{1\leq i \leq n}$, the map $\rho:M \to G$ in equation \EqRef{deff1} given in coordinates by  $x^i=f^i(p)$ for all $ p\in M$, satisfies equation \EqRef{pbf}. This follows at once by substituting $x^i=f^i$ in the expression for $\tau^i$ in equation \EqRef{Lieom}
which produces the expression for $\omega^i$ in equation \EqRef{MFP1}.
\end{proof}

\begin{Remark} \StTag{RM1} Theorem \StRef{CMF} shows that the for a solvable Lie group $G$ the map $\rho:M \to G$ can be found by quadrature and the matrix exponential and is in fact computed by the $n$ functions in Corollary \StRef{C2}.
\end{Remark}

\begin{Example} From Example \StRef{Ex1} we have the three forms $\omega^1,\omega^2,\omega^3$ on $M^0$ given in equation \EqRef{omex1} which satisfy,
$$
d\omega^1=- \omega^2 \wedge \omega^3, \quad d \omega^2 = 0, \quad d \omega^3 =0.
$$
The Lie group is $G=\real^3$ with coordinates $(x^1,x^2,x^3)$ and a basis of left invariant forms
\begin{equation}
\tau^1=dx^1+x^3 dx^2, \quad \tau^2 = dx^2,\quad \tau^3=dx^3.
\EqTag{mte1}
\end{equation}

The functions $f^1,f^2,f^3$ are given in equations \EqRef{om3} and \EqRef{ex1f1f2} in Example \StRef{Ex1} and these give the map $\rho:M^0 \to G$ in equation \EqRef{deff1}
 in Theorem \StRef{CMF} by
$$
\begin{aligned}
x^1 &= f^1= \frac{u_x^{10}+3u u_{xx}^2u_x^4+3 x u_x u_{xx}^3-3 u u_{xx}^3}{ 3 u_ x u_{xx}^3},  \\
x^2 &= f^2= \frac{u_x^6+2 u u_{xx}^2}{2 u_{xx}^2},\\
x^3 &= f^3= \frac{1}{u_x}-\frac{u_x^3}{u_{xx}}.
\end{aligned}
$$
It easy to check $\rho^* \tau^i=\omega^i$, $i=1,2,3$ with $\omega^i $ in \EqRef{omex1} and $\tau^i$ in equation \EqRef{mte1}
\end{Example}

Finally we now use our results from Section \StRef{Sreduce} along with Theorem \StRef{CMF} to show the following.

\begin{Theorem}\StTag{L3M}  Let $\lieg$ be an $n$-dimensional solvable Lie algebra and let $\beta=\{ {\bf e}_i \}_{1 \leq i \leq n}$ be a basis for $\lieg$ adapted to sequence of codimension one ideals $\liek_s\subset\lieg$, $s=0,\ldots,n$. Let $\bftau=[ \tau^1,\ldots,\tau^n]^T$ be the vector of left invariant one-forms on $G=\real^n$ with identity ${\bf 0}$,  defined in equation \EqRef{Lieom}.

On $G \times G$ with coordinates $(x^1,\ldots,x^n,y^1,\ldots, y^n)$ define the vector of $n$ differential one-forms $\bfomega=[\omega^1,\ldots,\omega^n]^T$ by
\begin{equation}
\bfomega = \rme^{-y^n [\text{\rm ad} ({\bf e_n})]}\cdots  \rme^{-y^1 [\text{\rm ad}( {\bf e_1})] }
 \pi_1^* \bftau + \pi_2^*\bftau
\EqTag{htheta}
\end{equation}
where  $\pi_1(x,y)=x$ and $\pi_2(x,y)=y$ are the two projection maps on $G\times G$.

\noindent
{\bf [i]} The forms $\omega^i$ satisfy
\begin{equation}
d \omega^i + \frac{1}{2} C^i_{jk} \omega^j \wedge \omega^k
\EqTag{dbfom}
\end{equation}
where $C^i_{jk}$ are the structure constants of $\lieg$ in the basis $\beta$.

\noindent
{\bf [ii]} Let $\mu:G \times G \to G$ with $\mu({\bf 0},{\bf 0})={\bf 0}$  be the map constructed in Theorem \StRef{CMF} so that $\mu^*\tau^i=\omega^i$. Then $\mu:G\times G\to G$ is the multiplication map for the Lie group $G$ with left invariant forms $\{ \tau^i \}_{1\leq i \leq n} $ and identity ${\bf 0} \in \real^n$.
\end{Theorem}

\begin{proof} The proof of this theorem relies on Corollary \StRef{CC1} below which in this case gives
$$
\Ad(\bfy^{-1}) = \rme^{-y^n [\text{\rm ad}( {\bf e_n})]}\cdots  \rme^{-y^1 [\text{\rm ad}( {\bf e_1})] },
$$
and so equation \EqRef{htheta} can be written
\begin{equation}
\bfomega =  [\Ad(\bfy^{-1})] \pi_1^* \bftau + \pi_2^*\bftau.
\EqTag{htheta2}
\end{equation}
Proposition 4.10 in \cite{sharpe:1997a} shows that $\omega^i =\mu^*\tau^i$ where $\mu:G\times G \to G$ is the
multiplication map. Therefore $\bfomega$ in \EqRef{htheta} satisfy the structure equations \EqRef{dbfom}.

To show part {\bf [ii]}, let $\rho:G\times G\to G$ be constructed by Theorem \StRef{CMF}, where $f^i$ are
chosen to satisfy $f^i({\bf 0},{\bf 0})= 0$. However as shown in Theorem 8.7 of \cite{sharpe:1997a} the multiplication map $\mu:G\times G \to G$ is the unique map satisfying $\mu^* \tau^i = \omega^i$ and $\mu({\bf 0},{\bf 0}) = {\bf 0}$. Therefore the function $\rho:G\times G \to G$  is the multiplication map.
\end{proof}

Note that Theorem \StRef{L3M} shows that the multiplication map $ \mu: G \times G \to G$ in Theorem \StRef{LG1}
is computed  by Corollary \StRef{C2}, and only involves $n$-quadratures and  the matrix exponential.  This is demonstrated in Example \StRef{Ex3} below. Also note that the identity element ${\bf 0}$ was chosen in Theorem \StRef{L3M} for simplicity.

The proof of Theorem \StRef{L3M} relies on Corollary \StRef{CC1} below. In order to prove Corollary \StRef{CC1} we present an alternative way to construct the multiplication map $\mu:G\times G \to G$ which is given in \cite{gardner:1989a}.

Let $\lieg$, $\beta$,  and $G=\real^n$ be as in Theorem \StRef{LG1}. On $G \times G$  define the rank $n$ Pfaffian system $I =\spn \{\theta^1,\ldots,\theta^n \}$ where
\begin{equation}
\theta^i =\pi_2^* \tau^i-\pi_1^*\tau^i
\EqTag{CTI}
\end{equation}
and $\tau^i$ are the left invariant forms on $G$ defined in equation \EqRef{Lieom}. The fact that the Pfaffian system is completely
integrable follows immediately from the structure equations for $\tau^i$.
Let $\CalL$ be the maximal integral manifold through $(x_0,y_0)$. It is not difficult to show that $\CalL$ is the
graph of the function $\psi:G \to G$ defined by
\begin{equation}
\psi(x) = \mu(y_0 x_0^{-1}, x)
\EqTag{prem}
\end{equation}
where $\mu:G \times G \to G$ is the multiplication map.

We now find an alternative set of generators for $I=\spn\{\theta^1,\ldots,\theta^n\}$ in equation \EqRef{CTI} which will allow us to find the maximal integral manifolds by quadratures which in turn determine  the function $\psi$ in equation \EqRef{prem} and hence the multiplication map.  The details are contained in the next two lemmas which also leads to the proof of Corollary \StRef{CC1}.

\begin{Lemma}\StTag{PreAd} Let $\lieg$ be an $n$-dimensional solvable Lie algebra and let $\beta=\{ {\bf e}_i \}_{1 \leq i \leq n}$ be a basis for $\lieg$ adapted to sequence of codimension one ideals $\liek_s\subset\lieg$, $s=0,\ldots,n$. Let $\bftau=[ \tau^1,\ldots,\tau^n ]^T$ be the vector of left invariant one-forms on $G=\real^n$  defined in equation \EqRef{Lieom}.  On $G\times G$ the $n$ differential one-forms
\begin{equation}
\tilde {\bftheta} = \rme^{x^1 [\text{\rm ad}(\be_1)]} \cdots \rme^{x^n [\text{\rm ad}(\be_n)]}
\left(\pi_2^* \bftau-\pi_1^*\bftau  \right)
\EqTag{bftf2}
\end{equation}
satisfy $I =\spn\{ \tilde \theta^1,\dots, \tilde \theta^n \}$, and
\begin{equation}
d \tilde \theta^i + \frac{1}{2} C^i_{jk} \tilde \theta^j \wedge \tilde \theta^k =0.
\EqTag{dtt0}
\end{equation}
\end{Lemma}

Given the generators $\bfttheta$ for $I$ in \EqRef{bftf2}, these can be used to construct the multiplication map as follows.

\begin{Lemma}\StTag{Pmu} With the hypothesis of Lemma \StRef{PreAd} on $\lieg$ $\beta$ and $\bftau$,  let $\rho: G\times G \to G$ be the unique map constructed in Theorem \StRef{CMF} so that
$\rho({\bf 0}, {\bf 0}) = {\bf 0}$ and
$$
\rho^*\tau^i = \tilde \theta^i
$$
where $\tilde \theta^i$ are given in equation \EqRef{bftf2}. Then
\begin{equation}
\rho(x,y) = \mu(y, x^{-1})
\EqTag{Fxy}
\end{equation}
where $\mu:G\times G\to G$ is the multiplication map with ${\bf 0}\in G$ as the identity.
\end{Lemma}
In particular Lemma \StRef{Pmu} allows us to determine the multiplication map $\mu$ for $G$  from $\rho$ using the inverse function theorem and solving
\begin{equation}
 z = \rho(x,y)
\EqTag{zxy}
\end{equation}
for $y$, giving from equation \EqRef{Fxy}, $y  = \mu(x,  z)$.

\begin{proof} (Lemma \StRef{PreAd}) On $G\times G$ consider the vector of $2n$ differential forms given by
\begin{equation}
\bfOmega=[ \theta^1, \ldots, \theta^n , \pi_1^* \tau^1, \ldots \pi_1^* \tau^n ]^T=\left[ \begin{array}{c} \bftheta \\ \bftau \end{array} \right]
\EqTag{dOm}
\end{equation}
where from equation  \EqRef{CTI} $\theta^i= \pi_2^* \tau^i-\pi_1^*\tau^i$.
 The structure equations for $\bfOmega$ follow from equation \EqRef{LTF} and are
\begin{equation}
\begin{aligned}
d \theta ^i +\frac{1}{2}C^i_{jk} \theta^j \wedge \theta^k + C^i_{jk} \theta^j \wedge \tau^k & =0, \\
d \tau^i  +\frac{1}{2}C^i_{jk} \tau^j \wedge \tau^k & = 0,
\end{aligned}
\EqTag{SE1}
\end{equation}
where we have dropped $\pi_1^*$. The $2n$ forms in \EqRef{dOm} are a basis of left invariant forms on $G \times G$ which are dual to the basis of left invariant vector fields $\lieg \times \lieg$ given by
\begin{equation}
\gamma= \{ \ X_1,\ldots, X_n,  X_1+Y_1, X_2+Y_2,\ldots , X_n+Y_n \ \}
\EqTag{BCD1}
\end{equation}
where we have written $X_i$ for the vector-field $X_i+0$ on $G\times G$.

Define subspaces $\liel_{s} \subset \lieg$, $s=0,\ldots, 2n$ by
\begin{equation}
\liel_{a}= \spn\{\ X_1,\ldots, X_n, X_1+Y_1, X_2+Y_2,\ldots , X_{n-a}+Y_{n-a}\ \}, \quad 0\leq a \leq n-1,
\EqTag{firstl}
\end{equation}
and
\begin{equation}
\liel_{n+a} = \spn\{\ X_1,\, X_2,\ldots ,\, X_{n-a} \ \}, \quad  0 \leq a \leq n.
\EqTag{secondl}
\end{equation}
It is then simple to show that $\liel_s\subset \lieg$, $s=0,\ldots,2n$ forms a sequence of codimension one ideals for the solvable Lie algebra $\lieg \times \lieg$. Furthermore, by definition, the basis \EqRef{BCD1} is adapted to the sequence of codimension one ideals $\liel_s\subset \lieg\times \lieg$.

We now apply Corollary \StRef{C1} using first $n$  ideals $\liel_{a}, a=0,\ldots, n$. This will result in a change in coframe on $G\times G$ where the terms $C^i_{jk} \theta^j \wedge \tau^k$ on the first line of equation \EqRef{SE1} don't appear (see paragraph 1 after the proof of Theorem \StRef{reduce}).

In order to apply Corollary \StRef{C1} we need to compute
$$
\text{\rm ad}_a (X_{n-a} +Y_{n-a}) : \liel_{a} \to \liel_{a}, \quad 0\leq a \leq n
$$
in terms of the basis \EqRef{BCD1}. This is easily done giving
$$
[ \text{\rm ad}_a (X_{n-a} +Y_{n-a}) ]=\left[ \begin{array}{cc} [\text{\rm ad}(\be_{n-a})] & 0 \\ 0 & [\text{\rm ad}_a(\be_{n-a})] \end{array} \right]
$$
where $[\text{\rm ad}_a(\be_{n-a})]$ is the $n-a$ by $n-a$
matrix representation of the derivation defined in equation \EqRef{alseq} in the basis $\beta=\{\be_i\}_{1\leq i \leq n}$ for $\lieg$, and we have used the isomorphism \EqRef{giso}.

If we consider the first matrix in equation \EqRef{seqreduce} we note that in $\Omega^{2n} = dx^n$ in equation \EqRef{dOm},
and the first reduction (which is equivalent to one application of Theorem \StRef{reduce}) with $f^n= x^n$ is given by
\begin{equation}
\hat \bfOmega =
\left[ \begin{array}{cc} \rme^{x^n[\text{\rm ad}(\be_n)]} & {\bf 0}_n  \\ {\bf 0}_n^T & \rme^{x^n [\text{\rm ad}(\be_n)]} \end{array} \right]
 \left[ \begin{array}{c} \bftheta \\ \bftau  \end{array} \right].
\EqTag{bfOm1}
\end{equation}
Now using equation \EqRef{Lieom} and the fact that $\rme^{x^n[\text{\rm ad}(\be_n)]} \rme^{-x^n[\text{\rm ad}(\be_n)]}  = {\rm I}_{n}$ equation \EqRef{bfOm1} simplifies to
\begin{equation}
\hat \bfOmega =
\left[ \begin{array}{c} \rme^{x^n[\text{\rm ad}(\be_n)]} \bftheta  \\ A_{n-1}(\bx) \ldots A_{2}(\bx) {\bf dx}.
\end{array} \right]
\EqTag{bfOm2}
\end{equation}
where
\begin{equation}
A_{n-a}(\bx) = \left[ \begin{array}{cc} \rme^{-x_{n-a}[\text{\rm ad}_a(\be_{n-a}]} & {\bf 0_a} \\ {\bf 0}^T_a & {\rm I}_{a} \end{array} \right], \quad a=0,\ldots, n-2.
\EqTag{PM}
\end{equation}
are the matrices in equation \EqRef{Lieom} and \EqRef{bfOm2}.

Continuing in this was we see that $f^i = x^i$ are the invariants and that after $n$ steps Corollary \StRef{C1} produces
$$
\left[ \begin{array}{cc} \rme^{x^1[\text{\rm ad}(\be_1)]} & {\bf 0}  \\ {\bf 0} ^T & {\rm I}_n \end{array} \right]
\left[ \begin{array}{cc} \rme^{x^2[\text{\rm ad}(\be_2)]} & {\bf 0}   \\ {\bf 0} ^T & A_2(-\bfx)  \end{array}\right]\cdots
\left[ \begin{array}{cc} \rme^{x^n[\text{\rm ad}(\be_n)]} & {\bf 0}  \\ {\bf 0} ^T & A_n(-\bfx) \end{array} \right]
\left[ \begin{array}{c} \bftheta \\ \bftau  \end{array} \right] =  \left[ \begin{array}{c} {\tilde \bftheta} \\ {\bf dx}  \end{array} \right].
$$
where again $A_s(\bfx)$ are given in \EqRef{PM} and $\tilde \bftheta$ are the sought after forms in equation \EqRef{bftf2}. By construction the structure equations  \EqRef{dtt0} hold which proves the lemma.
\end{proof}

We now turn to the proof of Lemma \StRef{Pmu}.

\begin{proof} (Lemma \StRef{Pmu}) We now apply Theorem \StRef{CMF}  to ${\bfttheta}$ in equation \EqRef{bftf2} which produces a function $\rho:G\times G \to G$ (using the matrix exponential and quadratures) such that
$$
\rho^* \tau^i =\tilde \theta^i
$$
and $\rho({\bf 0},{\bf 0})={\bf 0}$, where ${\bf 0}$ is the identity in $G$.
Let $(x_0,y_0)\in G \times G$, $z_0= \rho(x_0,y_0)$, and let
$$
\CalN = \{ (x,y) \in \real^n \times \real^n \ | \ \rho(x,y) =z_0 \}.
$$
is an $n$-dimensional integral manifold of $I$ through $(x_0,y_0)$ since  $\rho^1,\ldots,\rho^n$ are $n$ functionally independent first integral of $I$.
The maximal integral manifold $\CalL$ of $I$ through $(x_0,y_0)$ satisfies $\CalL \subset \CalN$ and  the projection map $\pi_1 : G \times G$ restricted to $\CalL$ is a diffeomorphism. Therefore $\pi_1:\CalN \to G$ is a covering map and hence a diffeomorphism.  Consequently $\CalN = \CalL$ and $\pi_2:\CalN \to G$ is also a diffeomorphism.

Now $\CalN$ is given by the graph in \EqRef{prem} and so given any $x\in G$ we can solve the equations $ \rho(x,y)=z_0$  for $y$   giving
\begin{equation}
y = g(z_0, x) = \psi(x)= \mu(y_0 x_0^{-1}, x).
\EqTag{detz}
\end{equation}
Therefore equation \EqRef{detz} implies $ z_0 = \mu( y_0, x_0^{-1}) $ and since $(x_0,y_0)$ was arbitrary this proves the theorem.
\end{proof}

\begin{Corollary}\StTag{CC1} Let $\lieg$ be a solvable Lie algebra with basis $\beta=\{ \be_i\}_{1\leq i\leq n}$ adapted to a sequence of codimension one ideals $\liek_s\subset \lieg,\ s=0,\ldots,n$, and let $G=\real^n$ be the Lie group in Theorem \StRef{L3M} with left-invariant forms $\{ \tau^i \}_{1\leq i \leq n}$ given in equation \EqRef{Lieom} in coordinates $(x^i)_{1\leq i \leq n}$ for $G$. Then
\begin{equation}
[\Ad(x)] =   \rme^{x^1 [\text{\rm ad} ({\bf e_1})] }\ldots\rme^{x^n [\text{\rm ad}( {\bf e_n})] }
\EqTag{Adx}
\end{equation}
where $[\Ad(x)]$ is the matrix representation of $\Ad(x),\ x \in G$ in the basis $\beta$.
\end{Corollary}

\begin{proof}  Let  $\rho :G \times G \to G$ be $\rho(x,y) = \mu(y, x^{-1})$. In components using the basis $\beta$ we have
\begin{equation}
\rho^* {\bftau} = [Ad(x)] \pi_2^* \bftau- \pi_1^* \bftau_R,
\EqTag{pbad}
\end{equation}
where $\bftau_R$ are the right invariant one forms on $G$ which equal the left invariant form $\bftau$ at the identity ${\bf 0}$.
From Lemma \StRef{Pmu} we also have  $\rho^* \bftau = \tilde \bftheta $. Using the expression in equation \EqRef{bftf2} for $\tilde \bftheta$ and equation \EqRef{pbad} gives
$$
\rme^{x^1 [\text{\rm ad}(\be_1)]} \cdots \rme^{x^n [\text{\rm ad}(\be_n)]}
\left(\pi_2^* \bftau-\pi_1^*\bftau  \right)=[Ad(x)] \pi_2^* \bftau- \pi_1^* \bftau_R
$$
leading to the expression for $[\Ad(x)]$ in equation \EqRef{Adx}.
\end{proof}

\begin{Example} \StTag{Ex3} We continue with Example \StRef{Ex2} and produce the multiplication map using Theorem \StRef{L3M} for the corresponding Lie groups where according to Theorem \StRef{LG1}, $\tau^i, i=1\ldots, 5$ in equation \EqRef{Ex1forms} form a basis for the left invariant forms. First we need the forms $\omega^i, \ i=1,\ldots,5$ in equation \EqRef{htheta} in Theorem \StRef{L3M}. We compute, using the structure constants in equation \EqRef{SCe1}, the matrix $\Ad(\bfy^{-1})$ in equation \EqRef{htheta} (or see equation \EqRef{htheta2}) to be
\begin{equation}
\begin{aligned}
&&& [\Ad(\bfy^{-1})]= \rme^{-y^5 [\text{\rm ad} ({\bf e_5})]}\cdots  \rme^{-y^1 [\text{\rm ad}( {\bf e_1})] } = \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \\
&&& \left[
\begin{array}{ccccc}{\rme}^{ a\, y^5 +b\, y^4 } & 0&0&-b \, y^1 {\rme}^{a\, y^5 + b\, y^4 }&-a\, y^1 {\rme}^{a\, y^5 a+b\, y^4}\\
0&{\rme}^{y^4} \cos y^5  &{\rme}^{y^4} \sin y^5 &-{\rme}^{y^4} (y^2 \cos y^5+y^3 \sin y^5 )&{\rme}^{y^4} (y^2\sin y^5-y^3 \cos y^5  )\\
0&-{\rme}^{y^4}\sin y^5 &{\rme}^{y^4} \cos y^5 &{\rme}^{y^4} (y^2 \sin y^5-y^3 \cos y^5 )&{\rme}^{y^4} (y^2 \cos y^5+y^3 \sin y^5 )\\
0&0&0&1&0 \\
0&0&0&0&1
\end{array} \right]
\end{aligned}
\EqTag{CFE}
\end{equation}
Now multiply  the vector of forms in equation \EqRef{Ex1forms} by the matrix in \EqRef{CFE} to produces the forms ${\bfomega}$ in equation \EqRef{htheta} as
\begin{equation}
\begin{aligned}
\omega^1& =\rme^{a\, y^5+ b \, y^4} \left( d y^1 +  \rme^{a\, x^5+ b \, x^4}dx^1  - a\,  y^1 d x^5 - b y^1 dx^4\right)\\
\omega^ 2& = \rme^{y^4}\left(  \rme^{x^4} ( \cos(x^5+y^5) dx^2+ \sin(x^5+y^5)dx^3) +\cos y^5  dy^2+\sin y^5 dy^3\right.\\
& \left. \qquad \qquad -(y^2\cos y^5 +y^3 \sin y^5)dx^4+
(y^2 \sin y^5 -y^3\cos y^5)dx^5 \right)\\
\omega ^3& =  \rme^{y^4}\left(  \rme^{x^4} ( \cos(x^5+y^5) dx^3-\sin(x^5+y^5)dx^2) +\cos y^5  dy^3-\sin y^5 dy^2\right.\\
& \left. \qquad \qquad
(y^2\sin y^5 -y^3 \cos y^5)dx^4+
(y^2 \cos y^5 +y^3\sin y^5)dx^5 \right)\\
\omega^ 4& =dx^4+dy^4 \\
\omega^5& = dx^5 + dy^5
\end{aligned}
\EqTag{MCEF}
\end{equation}
We now find, by quadratures, the functions in Corollary \StRef{C2} which satisfy $f^i({\bf 0},{\bf 0})={\bf 0}$ and as noted in Theorem \StRef{L3M}, this will produce our multiplication map $\mu$.

We find $f^4, f^5$ from $d\omega^4=0$ and $d\omega^5=0$ in the last two lines of equation \EqRef{MCEF} to be,
\begin{equation}
\begin{aligned}
f^5&=x^5 + y^5\\
f^4&=x^4 + y^4.
\end{aligned}
\EqTag{Ef45}
\end{equation}
Reducing  $\bfomega$ by using Theorem \StRef{reduce} two times applied to equation \EqRef{MCEF} we get,
\begin{equation}
\begin{aligned}
\hbfomega & = \left[ \begin{array}{cc} \rme^{(x^4+y^4 )[\text{\rm ad}_1(\be_4)]} & {\bf 0}_1 \\ {\bf 0}_1^T & 1 \end{array} \right] \rme^{(x^5+y^5) [\text{\rm ad}(\be_5)]} \bfomega \\
& =   \left[ \begin{array}{ccccc} \rme^{-b (x^4+y^4)} & 0 &  0 &0 & 0  \\ 0 & \rme^{-(x^4+y^4)} & 0 &0 & 0  \\ 0 & 0 & \rme^{-(x^4+y^4)} &0 & 0   \\ 0 & 0& 0 & 1 & 0 \\ 0 & 0 & 0 & 0& 1   \end{array} \right]
\left[ \begin{array}{ccccc} \rme^{-a (x^5+y^5)} & 0 &  0 & 0 & 0 \\ 0 & \cos (x^5+y^5)  &- \sin (x^5+y^5)  & 0& 0 \\ 0 & \sin (x^5+y^5)  & \cos (x^5+y^5) & 0  & 0\\ 0 & 0  &0 & 1 & 0\\ 0 & 0 & 0& 0 & 1\end{array} \right]
 \left[  \begin{array}{c} \omega^1 \\ \omega^2 \\ \omega^3 \\ \omega^4 \\ \omega^5 \end{array} \right] \\
& = \left[ \begin{array}{c}
dx^1+ \rme^{-a \, x^5-b\, x^4}\left(dy^1 -b\, y^1 dx^4-a \, y^1 dx^5\right) \\
dx^2+\rme^{-x^4}\left(\cos x^5 dy^2 -\sin x^5 dy^3-(y^2\cos x^5 -y^3\sin x^5 )dx^4-(y^3\cos x^5+y^2\sin x^5)dx^5\right) \\
dx^3+\rme^{-x^4}\left(\sin x^5 dy^2 +\cos x^5 dy^3-(y^3\cos x^5+y^2\sin x^5 )dx^4+(y^2\cos x^5-y^3\sin x^5 )dx^5\right) \\
dx^4+dy^4\\
dx^5+dy^5 \end{array} \right]
\end{aligned}
\EqTag{domh}
\end{equation}
where we have used  the matrix exponentials computed in equation \EqRef{Adms}.
Since $[\text{\rm ad}_3(\be_2)]$ $[\text{\rm ad}_2(\be_3)]$ are zero matrices equation there is no more reduction to be done in equation \EqRef{FIE} so we  have $d \homega^3=0, d\homega^2=0$ and $d \homega^1=0$ in equation \EqRef{domh}. This produces the final three functions ($\hat \omega^i=df^i,  f^i({\bf 0},{\bf 0})=0$) by quadratures,
\begin{equation}
\begin{aligned}
f^3 &=x^3 + \rme^{-x^4} (y^2 \sin x^5 +  y^3 \cos x^5), \\
f^2 & =x^2 + \rme^{-x^4}( y^2 \cos x^5 -y^3 \sin x^5 ),\\
f^1 &= x^1 + \rme^{-a\, x^5  - b\, x^4 } y^1 .
\end{aligned}
\EqTag{Ef123}
\end{equation}
Finally in accordance with Theorem \StRef{L3M} the multiplication map $z^i=f^i(x,y)$ is given using equations \EqRef{Ef45} and \EqRef{Ef123} by
\begin{equation}
\begin{aligned}
z^1 &= x^1 + \rme^{-a\, x^5  - b\, x^4 } y^1 ,\\
z^2 & =x^2 +\rme^{-x^4}( y^2 \cos x^5 -y^3 \sin x^5 ),\\
z^3 &=x^3 + \rme^{-x^4} (y^2 \sin x^5 +  y^3 \cos x^5), \\
z^4&=x^4 + y^4,\\
z^5&=x^5 + y^5.
\end{aligned}
\EqTag{MM}
\end{equation}
Equation \EqRef{MM} is the multiplication map ${ z}=\mu({ x}, {y})$ for the simply connected Lie group $G=\real^5$ with left invariant forms in \EqRef{MCEF} and ${\bf 0}$ as the identity.
\end{Example}

\section{Conclusion}   It seems possible that an explicit formula for the multiplication map $\mu$ in Theorem \StRef{L3M} can be given in terms of matrix exponentials but the author has not been able to find it. If this could be found then this would given an explicit formula for the multiplication map for any simply connected solvable Lie group given in closed form in terms of the structure constants.

\begin{bibdiv}
\begin{biblist}

\bib{anderson-fels:2005a}{article}{
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  author={Fels, M. E.},
  title={Exterior Differential Systems with Symmetry},
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  year={2005},
  volume={87},
  pages={3--31},
}

\bib{bouchetta:2005a}{book}{
author={M. Bouchetta, J. Morvan},
title={Differential Geometry and Topology Discrete and Computational Geometry},
publisher={IOS Press},
year={2005},
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\bib{doubrov:2000a}{article}{
author={B.M. Doubrov, B.P. Komrakov},
title={The constructive equivalence problem in differential geometry},
journal={Mat. Sb},
volume={191},
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pages={655--681}
}

\bib{fels:2007a}{article}{
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  journal ={Foundations of Computational Math.},
  volume = {7},
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  pages = {417-454}
}

\bib{fels:2008a}{article}{
  author={Fels, M. E.},
  title={Exterior Differential Systems with Symmetry},
  journal={The IMA Volumes in Mathematics and its Applications},
  year={2008},
  volume={144, II},
  pages={351--366},
  }

\bib{flanders:1989a}{book}{
author={H. Flanders},
title={Differential Forms with Applications to the Physical Sciences},
publisher = {Dover},
year={1963,1989}
}

\bib{gardner:1989a}{book}{
author={R.B. Gardner},
title={The Method of Equivalence and is Applications},
publisher = {SIAM},
year={1989}
}

\bib{griffiths:1974a}{article}{
  author={Griffiths, P.},
  title={On Cartan's method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry},
  journal={Duke Math. J.},
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  volume={41},
  pages={775--814}
}

\bib{jacobson:1962a}{book}{
author={N. Jacobson},
title={Lie Algebras},
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}

\bib{lychagin:1991a}{article}{
author={S.V. Duzhin},author={V.V. Lychagin},
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journal={Acta. Appl. Math.},
volume={24},
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\bib{olver:1998a}{book}{
author={Olver, P.J.},
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}

\bib{palais:1957a}{article}{
author={R.S. Palais},
title={A Global Formulation of the Lie Theory of Transformation Groups},
journal={Mem. Amer. Math. Soc},
volume={22},
year={1957},
}

\bib{sharpe:1997a}{book}{
  author = {Sharpe, R.W.},
   title = {Differential Geometry: Cartan's Generalization of Klein's Erlangen Program},
 publisher = {Springer},
  year={1997},
}

\bib{varadarajan:1984a}{book}{
  author = {V.S. Varadarajan},
   title = {Lie groups, Lie algebras, and their representations},
 publisher = {Springer},
  year={1984},
}

\bib{winternitz:1976a}{article}{
author={J. Patera}, author={R. T. Sharp}, author={P. Winternitz}, author={H. Zassenhaus},
title={Invariants of real low dimensional Lie algebras},
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volume={17}, number={6},
year={1976},pages={966--994}
}

\end{biblist}
\end{bibdiv}

\end{document}

redpajama_set_name: RedPajamaArXiv


text:
Home / Blog / UK and Ireland: COVID-19 Legislation and Guidance Update
UK and Ireland: COVID-19 Legislation and Guidance Update
September 18, 2020 Health & Safety, ISO Standards, Legislation Update
(This blog post covers some of predominate legislation & guidance between 18th July – 15 September 2020)
The Planning and Development Act 2000 (Exempted Development) (No.3) Regulations 2020 (SI No. 293 of 2020) apply from 11 August 2020 and cease to have effect on the day on which Part 3 of the Health (Preservation and Protection and other Emergency Measures in the Public Interest) Act 2020 (No. 1 of 2020) ceases to have effect. The Regulations provide for construction, erection or alteration of temporary awnings, marquees, portable cabins or covered pedestrian or set down areas connected with the management of airport passenger movement in a State airport for purposes connected with the prevention of transmission of Covid-19.
The Health Act 1947 (Section 31A – Temporary Restrictions) (Covid-19) (Face Coverings in Certain Premises and Businesses) Regulations 2020 (SI No. 296 No. 2020) require the wearing of face coverings in certain premises. A person shall not, without reasonable excuse, enter or remain in premises where goods are sold directly to the public or premises\ set out in the Schedule to the Regulations, without wearing a face covering. The Regulations do not apply to children aged under 13 years.
The Road Traffic (Licensing of Drivers) (Amendment) (No. 7) Regulations 2020 (SI No. 299 of 2020) amend the Road Traffic (Licensing of Drivers) Regulations 2006 (SI No. 537 of 2006) to provide an extension to the validity of certain driving licences during the Covid-19 pandemic.
Driving licences that expired or are due to expire between 1 March 2020 and 31 August 2020 have been extended for 7 months. The date of expiry for these licences shall be 7 months after the expiry date specified on the licence.
The Financial Provisions (Covid-19) (No. 2) Act 2020 (No. 8 of 2020), introduced on 1 August 2020, provides for the imposition, repeal, remission, alteration and regulation of taxation and make further provision in connection with finance. It amends the Emergency Measures in the Public Interest (Covid-19) Act 2020 which introduced the Temporary Wage Subsidy Scheme (TWSS). This has been replaced by the Employment Wage Subsidy Scheme (EWSS) from 31 August 2020. Furthermore, Revenue has published "Guidelines on the Operation of the Employment Wage Subsidy Scheme".
The Environmental Assessment of Plans and Programmes (Coronavirus) (Amendment) Regulations 2020 (SI 2020 No. 734) amend the Environmental Assessment of Plans and Programmes Regulations 2004 ("the SEA Regulations"), which apply in England, on a temporary basis from 16th July 2020 until 31st December 2020. The Regulations modify requirements placed on responsible authorities and the Secretary of State for documents relating to strategic environmental assessment to be available for physical inspection by members of the public at an address, for consultees to be informed of that address and for a copy of those documents to be available to be obtained from that address.
The Department for Environment, Food & Rural Affairs has published guidance on "Coronavirus (COVID-19): disposing of waste". This guidance provides information on how to dispose of personal or business waste, including face coverings and personal protective equipment (PPE), during the coronavirus pandemic.
The Business and Planning Act 2020 (2020 c. 16) supports economic recovery and growth as the UK recovers from the disruption caused by COVID-19. The Act includes a range of measures to help businesses adjust to new ways of working as the country recovers from disruption caused by Covid-19. In relation to planning, the Act makes temporary changes to the law in England, as well as new, permanent provision for certain planning proceedings in England to be considered by means of more than one procedure. This will ensure that the planning system can continue to operate effectively and support the planning and safe construction of new development following the impact of Covid-19.
The Corporate Insolvency and Governance Act 2020 (2020 c. 12) make provision about companies and other entities in financial difficulty and includes temporary measures aimed at easing some of the most pressing consequences businesses may be experiencing as a result of the coronavirus (COVID-19) pandemic.. The Act makes temporary changes to the law relating to the governance and regulation of companies and other entities.
The Health Protection (Coronavirus, Wearing of Face Coverings in a Relevant Place) (England) Regulations 2020 (SI 2020 No. 791) require members of the public to wear face coverings, from 24 July 2020, while inside shops, shopping centres and transports hubs in England, to protect against the risks to public health arising from coronavirus, except in certain limited cases. The Health Protection (Coronavirus, Wearing of Face Coverings in a Relevant Place) (England) (Amendment) Regulations 2020 (SI 2020 No. 839) require members of the public to wear face coverings in additional indoor premises in England, from 8 August 2020.
The Health Protection (Coronavirus, Wearing of Face Coverings) Regulations (Northern Ireland) 2020 (SR 2020 No. 151) require members of the public, subject to limited exceptions, to wear face coverings on public transport to protect against the risks to public health arising from coronavirus. The Health Protection (Coronavirus, Wearing of Face Coverings) (Amendment) Regulations (Northern Ireland) 2020 (SR 2020 No. 164) require members of the public to wear face coverings whilst inside shops and enclosed shopping centres in order to protect against the risks to public health arising from coronavirus, except in certain limited cases.
The Health Protection (Coronavirus) (Restrictions and Requirements) (Scotland) Regulations 2020 (SSI 2020 No. 279) adjust the restriction on parties in private dwellings to ensure student accommodation is appropriately covered by the existing restrictions. The Regulations introduce new restrictions on public and private gatherings, setting out that gatherings of no more than 6 people from no more than 2 households are restricted, with the exception of some limited circumstances. The Regulations also adjust the previous provisions relating to face coverings to ensure that customers and staff are required to wear face coverings, in certain circumstances, within a hospitality venue.
Coronavirus , COVID-19 , Health & Safety , legislation update , Legislation Updates Ireland , legislation updates UK
Returning to the Office WRC Code of Practice for Employers and Employees on the Right to Disconnect Risk Assessments for COVID-19: your responsibilities as a company Your responsibilities for remote workers The importance of office ventilation during the Covid-19 epidemic UK and Ireland: COVID-19 – Legislation and Guidance Update New Biological Agents Regulations and Code of Practice 2020 COVID-19 Legislation and Guidance Update October 2020 Autumn Legislative Programme Published The Pandemic and Our Environment

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THE London West End premiere of a Wirral musical about First World War poet Wilfred Owen has sold out.
Bullets And Daffodils tells the story of his short time in the borough and even contains a song called Welcome To Birkenhead.
Created by Oxton musician Dean Johnson, it will be staged at the Jermyn Street Theatre near Covent Garden this Sunday, July 29.
Owen's iconic poems are set to music in the production, which features actor Christopher Timothy, who played vet James Herriot in BBC drama All Creatures Great And Small, and Graeme Clark from 80s band Wet Wet Wet.
Also in the cast is Chloe Torpey, a rising star of Wirral's acting scene who has been described as the next Glenda Jackson.
Dean, a lifelong fan of Wilfred's work, said: "The first performance of Bullets And Daffodils was at the Birkenhead Central Library just under two years ago.
The play explores Wilfred's relationship with his mother through his letters, poetry and his life's events.
It has two characters, Wilfred and his mother Susan, and is set in a Victorian sitting room.
The dialogue is interspersed with songs written and performed by Dean. The play has the full support of the Wilfred Owen Association.
Looking forward to taking the show to the West End, Dean said recently: "It's development has been really fast and I'm delighted that we've made it to London.
"Jermyn Street is quite an intimate theatre, which is very well known for staging premieres, so to have the chance to appear there is a huge honour."
The original production premiered in Birkenhead Central Library in 2010 and has since been staged at a number of Wirral venues.
A host of well known actors including ex-Brookside actor Dean Sullivan and Hoylake-based writer/producer John Gorman have appeared in the show.
But, as Dean Johnson explained, the West End show promises some changes. He said: "The production is almost unrecognisable from how it was in the early days."
Born in Oswestry in 1893, Wilfred was brought up in Birkenhead and is recognised as one of the greatest voices of the First World War.
He also has a road named after him, on the former site of Birkenhead Institute, which is now a housing estate.
In 1915, he enlisted in the British Army and was killed, aged 25, on November 4, 1918, during the battle to cross the Sambre-Oise canal at Ors in Northern France.
At the time of his death he was virtually unknown. Only four of his poems were published during his lifetime.
But he had always been determined to be a poet and had experimented with verse from an early age.
Among his 62 poems are 1914, Dulce et decorum est and Anthem For Doomed Youth.
The next local performance of the musical is at The Heswall Festival in October.


These paintings are lartger than I have done for a while 24"x36" and 2" deep so that they can be hung without frames. I intend to do two more 'Summer' and 'Spring'.
As in the "Fall Series" I've taken my Plein Air attempts and used them as notes that I expand upon using the technique I've developed resembling a semi abstract perhaps 'tonalist' rendering of a landscape subject. I've discovered that my affinity for color and pushing paint around has developed into a distinct technique that uses something of the effect of Seurat's 'pointilist' practice of placing pure color in proximity to each other to create a third color. The technique achieves a vibrancy but working spontaneously and expressively, mixing color on the canvas, I have developed a personal application of paint through brushstrokes that imparts a vigorous and I think, 'unique' effect that allows the canvas to become less the rendering of an image or a picture of something as an artifact in it's own right.
As each picture developes I discover new things that I wish to pursue in the next painting. Working in series helps develop both a theme and continuity of process.
In an effort to apply to shows and contests I made a CD of my work and was disconcerted to find that although there are many pieces that I liked and had good things going on, I realized that what the public or 'industry' would be looking for or more comfortable with, had to have continuity.
I think I am achieving a way of working that I am comfortable with and one that I will want to work with for some time to come as it has much potential and is natural for me yet challenging in that it has intellectual prospects that can disclose both subjective and objective issues.


SAN PEDRO CITY— A 14-year-old girl was found dead in Alaminos town in Laguna province early on Sunday.
Supt. Chitadel Gaoiran, spokesperson of the Cavite, Laguna, Batangas, Rizal, Quezon police, said in a text message that the victim, whose name was withheld, was possibly raped.
Circumstances that led to the assault were not immediately disclosed, although Gaoiran said the body was recovered in a vacant lot in Barangay San Gregorio past midnight on Sunday.


History: Every year additional dogwoods have expanded the collection to over 30 different types of dogwoods in the collection as of Spring 2014.
Cherokee Brave, a nice pink flowered form of our native dogwood, is putting on a show this spring.
The Bob Timberlake dogwood, a double flowered form of our native dogwood, is heavy with flower this spring.


text:
In 2013, we began a $10,000,000 capital campaign to celebrate Prep's 50th Anniversary as a school. This past week, Campaign Chair Carol Romero-Wirth, announced victory with close to $15,000,000 raised for our community.
We made it Santa Fe!
Eighty percent of the money raised as part of the "Generating Impact: Prep Beyond 50" campaign will provide exceptional educational opportunities for our state's brightest students with financial need–more than $12,000,000 will provide tuition assistance for these talented students attending Santa Fe Prep and Breakthrough Santa Fe.
Included in our numbers is a phenomenal donation from trustee Andrew Davis, whose generosity is truly changing New Mexico. The Davis New Mexico Scholarship will give millions annually to provide college scholarships to first-generation and low-income students from Northern New Mexico. In this inaugural 2016 year, eighteen qualified students from Santa Fe and Albuquerque were awarded full scholarships to attend one of five Davis New Mexico partner colleges. Eventually, the program will grow to support 25 students each year.
The 50th Anniversary campaign also funded the beautiful Sun Mountain field for our athletes, a renovation of Santa Fe Prep's middle and upper schools, and the creation of the E.E. Ford Teachers' Colloquium, which provides summer inspiration and rejuvenation for Santa Fe public school teachers for free and independent school teachers from around the country for a small fee.
Our 50th Anniversary campaign lived up to its name by generating impact throughout Santa Fe and northern New Mexico and changing lives along the way. We are exceptionally grateful to Campaign Chair Carol Romero-Wirth, our Steering Committee, Board Chair Kelsey Daly Brown, and to our donors who made this campaign's success possible. We are launching into the next 50 years with plans to continue our mission as a private school with a public purpose.
The Advancement Office
Categories: Featured Stories


text:
Q: How to put a complete dark theme into my UWP? I'm trying to made an UWP app that can change of theme. I'm using the instruction RequestedTheme = ElementTheme.Dark;, but this only changes the "basic items", I'm talking about the background and text.
The textBox, the ContentDialog and others aren't into the dark mode.
I'd like to find a way to "dark" ALL the items into the app, not only the text and background.
Hope there's any way.

A: You can set theme on the application and element level.
If you want to set it on application level, it must be done when the app is started (in App.xaml.cs constructor). This setting cannot be changed at runtime.
To switch theme at runtime, you can utilize the RequestedTheme property on a control. This applies the theme to the control and its children in the visual hierarchy. This can be switched any time at runtime. To make the selected theme apply to the whole application, you need to set it on a top-level visual element - ideally the root element of the window:
var root = (FrameworkElement)Window.Current.Content;
root.RequestedTheme = ElementTheme.Dark;

Usually, the window's content is a Frame but, more generally it can be an arbitrary FrameworkElement.

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text:
Jeannette A. Geib
There are condolences waiting approval on Jeannette's Tribute wall
Plant a tree in memory of Jeannette
Celebrating the life of Jeannette A. Geib
Be the first to share your favorite memory, photo or story of Jeannette. This memorial page is dedicated for family, friends and future generations to celebrate the life of their loved one.
Obituary for Jeannette A. Geib
Jeannette A. Geib, 89 ,
Former Willimantic resident, died Dec. 19, 2017, at Hartford Hospital. At her request, no service will be held.
Mrs. Geib was born June 16, 1928, Norwich, Conn., to Cyril and Maryanne McClure. Her final days were spent surrounded by family and friends that Jeannette had touched in some small way throughout her very memorable life.
Jeannette is survived by her brother, Harvey McClure/FL. and her sister, Irene Ouellette, also two daughters, Diane Schreiner and Ann M. Eichner and one Son, Louis P. Dumont Jr./FL.
We will miss her bright smile and candid wit
Love you Mom.
Her family will receive relatives and friends Wednesday, 12/27/17, from 6:00 to 8:00 p.m. at the Bacon Funeral Home, 71 Prospect St. Willimantic, CT. On December 28, 2017 at 10:00 a.m. a prayer will be said for Jeannette at the funeral home followed by a procession to the New Willimantic Cemetery on Route 32 in Willimantic for a Graveside Service.
To plant a tree in memory of Jeannette A. Geib, visit the Tribute Store.
Welcome to Jeannette's Memorial Wall
Share Jeannette A. Geib's life celebration
"Email Address" would like to share the life celebration of Jeannette A. Geib. Click on the "link" to go to share a favorite memory or leave a condolence message for the family.


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New, Anniversary Series, Vol. 14: The Most Beautiful Concert Highlights from Maulbronn Monastery, 2014-2017 (Live)
Michael Martin Kofler (flute), Sarah Wegener (soprano), David Allsopp (counter-tenor), Christoph Soldan (piano), Birgit Schnurpfeil (violin), Julia Schröder (violin), Matthias Hummel (violin)
Maulbronn Chamber Choir, Timo Handschuh, Jürgen Budday
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Stamitz, C: Symphony in E-Flat Major, Op. 13 No. 1: II. Andante moderato [Live]
Südwestdeutsches Kammerorchester Pforzheim
Timo Handschuh
Symphony in E-Flat Major, Op. 13 No. 1, DTB VIII / 2.3: II. Andante non moderato (Live)
Mozart: Flute Concerto No. 1 in G Major, K. 313: I. Allegro maestoso [Live]
Michael Martin Kofler (flute)
Mozart: Symphony No. 21 in A Major, K. 134: I. Allegro [Live]
Mozart: Symphony No. 21 in A Major, K. 134: II. Andante [Live]
trad.: Auld Lang Syne [Live]
Ensemble il capriccio
Sarah Wegener (soprano)
Handel: Rinaldo, HWV 7a: Lascia ch'io pianga [Live]
Ferrandini: Giunta l'ora fatal (Il pianto di Maria): Se d'un Dio [Attrib. G.F. Handel] [Live]
Il pianto di Maria: II. Se d'un Dio (Live)
Handel: Gloria (Excerpts)
Gloria (Excerpts): I. Gloria in excelsis Deo [Live]
Gloria (Excerpts): II. Et in terra Pax [Live]
Gloria (Excerpts): VI. Quoniam tu solus sanctus [Live]
Bach, J S: Johannespassion, BWV 245 (Excerpts)
Maulbronn Chamber Choir, Ensemble il capriccio
Jürgen Budday
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St. John Passion, BWV 245, Pt. 1 (1724 Version Excerpts): No. 1, Herr, unser Herrscher [Live]
St. John Passion, BWV 245, Pt. 1 (1724 Version Excerpts): No. 3, O große Lieb [Live]
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Show 5 remaining tracks for Bach, J S: Johannespassion, BWV 245 (Excerpts)
St. John Passion, BWV 245, Pt. 1 (1724 Version Excerpts): No. 7, Von den Stricken meiner Sünden [Live]
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St. John Passion, BWV 245, Pt. 2 (1724 Version Excerpts): No. 15, Christus, der uns selig macht [Live]
St. John Passion, BWV 245, Pt. 2 (1724 Version Excerpts): No. 39, Ruht wohl, ihr heiligen Gebeine [Live]
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Mozart: Piano Concerto No. 11 in F Major, K. 413 (Arr. for Piano & String Quintet): I. Allegro [Live]
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Christoph Soldan (piano)
Mozart: Piano Concerto No. 11 in F Major, K. 413 (Arr. for Piano & String Quintet): II. Larghetto [Live]
Mozart: Piano Concerto No. 12 in A Major, K. 414 (Arr. for Piano & String Quintet): I. Allegro [Live]
Bach, J S: Concerto for 2 Violins in D Minor, BWV 1043: I. Vivace [Live]
Lautten Compagney
Birgit Schnurpfeil (violin), Julia Schröder (violin)
Vivaldi: Concerto for 4 Violins in B Minor, Op. 3 No. 10, RV 580: I. Allegro [Live]
Julia Schröder (violin), Birgit Schnurpfeil (violin), Matthias Hummel (violin)
Vivaldi: Concerto, Op. 3 No. 11 'Con due Violini e Violoncello obligato', RV 565
Birgit Schnurpfeil (violin), Matthias Hummel (violin)
Concerto grosso in D Minor, Op. 3 No. 11, RV 565: I. Allegro - II. Adagio e spiccato - III. Allegro (Live)
Concerto grosso in D Minor, Op. 3 No. 11, RV 565: IV. Largo e spiccato (Live)
Concerto grosso in D Minor, Op. 3 No. 11, RV 565: V. Allegro (Live)
Bach, Johann Sebastian (1685-1750)
Ferrandini, Giovanni Battista (1709-91)
Handel, George Frideric (1685-1759)
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text:
Over The Rainbow & In A Garden
June 30, 2015 July 11, 2015 / Kevin
There are times when a gardener has to step off the garden path, when he or she has to set foot on mulch and tip-toe further in to investigate new growth, a weed, or a pest. Sometimes, it's just to get a new perspective.
It's the same thing with bloggers. Sometimes, you just have to step away from your theme — and for this post, I'm stepping off the garden path.
It's been an amazing few days. My marriage is now recognized from sea to shining sea. My head is still reeling from the historic significance of the Supreme Court's controversial decision, and to help regain focus, I visited a local community garden, where people from different backgrounds come together to learn, to share, to grow.
When the Court issued its decision, Joe and I were about to leave the house. We heard the news, applauded, and went about our day. I first dropped Joe off for a colonoscopy and then went food shopping. That's marriage.
It wasn't until we were both home and going through our personal FaceBook feeds that the scope and impact of the decision really hit us. As rainbow after rainbow appeared on the profile photos of straight and gay friends, as street celebrations broke out across cities and towns, and as landmarks across the nation were illuminated in rainbow colors, I cried. I was over the rainbow.
The dreams I had dared to dream really did come true . . . In my lifetime . . .
This was a far cry from the '60s when bars were constantly raided, when homosexuality was a psychiatric diagnosis; a far cry from the '80s, when I was a young gay and AIDS was in every headline; a far cry from just five years ago.
Amid all of these celebrations, there was sadness. Just prior to the Court's decision, there was again unspeakable violence, this time in a church in Charleston. Memorials, again. Prayers, again. A call for gun control, again. A caution to not jump to conclusions, again. Empty words, again. Racism, again.
And a call for the removal of the Confederate flag.
As Confederate flags came down and rainbow flags went up, I engaged in a FaceBook tug-of-war with a friend who believes in traditional marriage. He was not happy with the Court's decision. Period.
I tried to reason with him, to explain to him that under the Court's ruling he is allowed to retain his faith, that his life will not change, and that if he doesn't believe in same-sex marriage, he shouldn't marry a man.
Seeds come in all shapes and sizes — and these seeds of understanding and acceptance would not take hold in his hardscrabble mind. He would not — could not — budge from his position.
Although he never said it, his words meant that my near 30-year relationship with Joe — no matter how much equality I had just received courtesy of the Supreme Court — was and will always be in his eyes less than his own marriage.
And that's when all of it — the rainbows and stars and bars — all blurred and all clicked.
Despite the Court's decision and despite the removal of the Confederate flag, there is still so much work to be done, so many conversations to have regarding race and sexuality and gender and, well, everything.
It's like this community garden.
Just because there are raised beds of tomatoes and herbs, swiss chard and sunflowers, the garden still needs tending — just as a home garden does.
There's weeding and fertilizing and vigilant pest removal. There's watering and thinning and staking. There's minding your own plot so that any problems don't jump to other plots, so that a minor problem doesn't become an infestation.
In a community garden, though, there's also the tending of people: sharing, negotiating, cooperating, understanding, accepting, educating, changing, and growing.
In short, maintaining a community garden takes a lot of work.
You might even say it takes a village.
Inspiration, Nitty Gritty Dirt Man
← A Tale Of Two Tomatoes
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22 thoughts on "Over The Rainbow & In A Garden"
Well said. It was a big week but much work remains.
Absolutely, PBM. Absolutely.
Pat Loud
Well said, dear Kevin! I celebrate the Court's decision with you and Joe and my heart breaks that there are those individuals who may never agree with it or accept your marriage. What they don't seem to understand is that your union is so much stronger than most, in part because you guys had to fight so hard and wait so long to make it legal and accepted. I often brag to people about how happy you are and what a loving relationship you have – one that is so much better than my 'traditional' marriage ever was – because you don't take for granted what you have! I love you and Joe so much and cried with joy when I heard the news! We should all be sowing seeds of knowledge and understanding and love, and maybe, someday, we can overcome hatred and prejudice. What a day that will be!
Hi Aunt Pat. Thank you for your words and love and support. It seems that as a nation we are becoming more and more polarized. Rather than a melting pot or a mosaic, we seem — and sadly act — like a shattered piece of ceramic. Still, we should all work toward that "someday." Love ya.
Cathey Harper
As I recall, the first rainbow came after global catastrophe. Ever since, they appear after storms. Unfortunately, there seems a sorrow based cause for the rainbow's triumph, be it in meteorological or societal form. But I joy in this rainbow's arrival, too. Hugs across cyberspace.
Hello dear friend. Beautifully said. 🙂
Kat at travelgardeneat
There's a reason our country was founded on a principle of separation of church and state ….. so many discussions in recent days highlight the importance of it. There is still much work to be done on many fronts, and will always be. Cheers to you and Joe and for many rainbow-filled years ahead! Lovely post, Kevin.
Hi Kat. I agree. It's always difficult to step into a controversial topic in a blog, but I tend to think we all have more in common that we have differences. I think it all could be settled with calm and honest discussion and a delicious meal. I think we have to be open to hear one another. Be well — and Happy 4th!
Demelza Poldark
Congratulations on being together for almost 30 years. Many marriages have failed long before that.
The Supreme Court decision is just one step, with many more to come. I think we are a few steps ahead of you here in Canada, but we also have a long way to go.
Great blog post today.
Hi Demelza. Canada is way ahead! That being said, though, there are probably times when nations experience growing pains — this is one of those times. I think the hard part is keeping our rage against one another in check so that we can listen. Thanks for commenting.
Congratulations, Kevin and Joe! And although I'm sorry about the rift with your friend, sometimes there's simply nothing we can do or say that will change someone's perspective — they are convinced of the "truth" or "rightness" of their mindset and simply refuse to concede that someone else's might have a shred of validity. But that problem is theirs, not yours. 🙂
Hello Chatsworth. Well said. In truth, my friend is a FB friend — an odd relationship in this day and age. My face-to-face friends say I should unfriend him, but that seems to easy. I think there needs to be constant dialogue and understanding and empathy. We lose all that in the yelling. Happy 4th!
Congratulations to you and Joe on your newly legitimated status! Imagine, you can go on vacation anywhere in the country and know that your marriage will be recognized! The court decision isn't a panacea and it doesn't instantly change people's minds, but it's an important step. It has been wonderful to see all the photos this week of elderly couples (including 2 women who have been together for 73 years!) getting married.
Hi Jean. It has been all so very emotion for both sides. I often think that if opposing sides could just hang out with one another — forget the divisive issue — but just see each other as people. I recently heard of a story of two men who met during a war. They were enemies and found themselves seeking shelter in the same place — each instructed to kill the other. One man, though, was injured — and his enemy went through his pockets, he found a photo of his family. The enemy became human, and so he helped him to survive. After the war, each man had emigrated from their respective countries — and by chance, they reunited in the lobby of a clinic that specialized in PTSD in their new country. Thanks for your kind words and Happy 4th!
That ruling was indeed a triumph! But reading the news articles made me feel…uneasy. There is an incredible amount of power invested in a group of nine unelected people.The rights of a large minority of Americans really came down to only one guy. And Alito's statements in the dissenting opinion were blatantly political and religious – a tacit admission that the four dissenters could find no support for their opinion in the actual Constitution. Scary.
Hi Ann. Clearly, I agree and am thrilled with the decision — but it does pull to the forefront quite a slippery slope the separation of church and state. Since its birth, the United States has been a nation like no other — a hodgepodge of people who came together to make something, to have to exist beside one another. On paper, it sounds lovely. In reality, it takes work — and as we have become a hyphenated nation (_______-American), we seem to be retreating into our tribes. Still, I think we have more in common with one another. We just have to be willing to listen and hear, not half listen and yell. Thanks for your thoughts — and have a Happy 4th!
kkeevins
Ignorance is NOT bliss, but it certainly is alive and well, isn't it?
I share your tears of joy and your tears of sadness.
Bravo to the Supreme Court — finally!
Bravo to you and Joe, Kevin!
Love, —K
Hi K. Thank you for your kind words and support in everything. You know, this blog wouldn't even exist if it weren't for you. You planted that seed! Have a wonderful 4th of July!
You're a sweetheart, Kevin. You just made my day. Have a wonderful 4th!
I'm so glad I didn't miss this post, Kevin. You've shared a very personal perspective with wonderful clarity. We were out of town when the Supreme Court decision came through and I sort of did the opposite of you–I didn't go on FB. I was so afraid I'd start reading comments and remarks that would be running much closer to your friend's strongly held view of traditional marriage, and I just didn't want to be confronted with that when I was so thrilled. I do agree with you that there is still a lot of work to be done to ensure acceptance of the Court's decision. Someone I love very much is 81 years old and considering marrying her partner of more than 40 years. How can I be anything but thrilled. When I think what she has experienced and lived over 80 years, how thankful I am she lived to see this day. Your words have touched me very much, Kevin. And I hope you and Joe continue to celebrate this victory…it's a BIG one. 🙂
Hi Debra. I love your friend's story — and what a wonderful post for you to share, to record her perspective on what she's been able to see in her life. It's tremendous — and I'm always flabbergasted that there are people who cannot see the beauty of that relationship, but also the strength in two people who have remained together in the face of obstacles. That's love, and that's what it's all about. Be well!


text:
The present state of England in regard to agriculture, trade, and finance : with a comparison of the prospects of England and France
Joseph Lowe
The publication "The present state of England in regard to agriculture, trade, and finance : with a comparison of the prospects of England and France" is placed in the Top 100 in 1824.
ci.nii.ac.jp/ncid/BA63615251
Joseph Lowe, The present state of England in regard to agriculture, trade, and finance : with a comparison of the prospects of England and France (1824)
Joseph Lowe, The present state of England in regard to agriculture, trade, and finance : with a comparison of the prospects of England and France (1824)
{{cite web | title=The present state of England in regard to agriculture, trade, and finance : with a comparison of the prospects of England and France | year=1824 | url=https://citeweb.info/18240000010 | author=Joseph Lowe}}


text:
O barão Heinrich von Berenberg-Gossler (1907–1997) era um banqueiro alemão, membro da ilustre dinastia bancária Berenberg-Gossler e proprietário e chefe do Berenberg Bank.

Ele era filho do barão Cornelius von Berenberg-Gossler e Nadia, née Oesterreich (1887–1962) e sobrinho do senador e embaixador John von Berenberg-Gossler.

Na década de 1930, trabalhou no Banco de Londres e América do Sul em Buenos Aires. Tornou-se sócio do Berenberg Bank em 1935 e, posteriormente, presidente. Ele também foi cônsul geral de Mônaco.

Literatura

 Manual Genealógico dos Adels, Banda 16, Freiherrliche Häuser B II, CA Starke Verlag, Limburgo (Lahn) 1957

Banqueiros da Alemanha

redpajama_set_name: RedPajamaWikipedia