Let $\mathrm{\𝔤}\mathit{ }\mathit{ }$be a Lie algebra and $\mathrm{\𝔥}$ an ideal in $\mathrm{\𝔤}$.  Then elements of the quotient algebra $\mathit{ }\mathrm{\𝔤}\mathit{\/}\mathrm{\𝔥}$$$are the cosets $x \+\mathrm{\𝔥}\mathit{ }$, where $x\mathit{ }\in \mathrm{\𝔤}$.  The Lie bracket on $\mathrm{\𝔤}\mathit{\/}\mathrm{\𝔥}$ is defined by $\left[x \+\mathrm{\𝔥}\mathit{\,}\mathit{ }y\mathit{ }\mathit{\+}\mathit{ }\mathrm{\𝔥}\right] \= \left[x\, y\right] \+ \mathrm{\𝔥}$. If vectors $\left\{y_1\, y_2\, ..\. \,y_r\right\}$ form a basis for a complement to $\mathrm{\𝔥}$, then the cosets $\left\{y_1\+ \mathrm{\𝔥}\, y_2 \+ \mathrm{\𝔥}\, ..\. \,y_r \+\mathrm{\𝔥}\right\}$ form a basis for $\mathrm{\𝔤}\/\mathrm{\𝔥}$.

The program QuotientAlgebra(h, m) creates a Lie algebra data structure for the quotient algebra $\mathrm{\𝔤}\/\mathrm{\𝔥}$. using the vectors in the complement m as the representative basis elements for $\mathrm{\𝔤}\/\mathrm{\𝔥}$.

A Lie algebra data structure contains the structure constants in a standard format used by the LieAlgebras package (see LieAlgebraData). The command DGsetup is then used to initialize a Lie algebra -- that is, to define the basis elements for the Lie algebra and its dual and to store the structure constants for the Lie algebra in memory.

With the optional keyword present, QuotientAlgebra(h, m, "Matrix") returns the Lie algebra data structure for $\mathrm{\𝔤}\/\mathrm{\𝔥}$ and the matrix representation of the canonical projection map $\mathrm{\π} \: \mathrm{\𝔤}\mathit{ }\mathit{\to }$$\mathrm{\𝔤}\mathit{\/}\mathrm{\𝔥}$ defined by $x\to x \+ \mathrm{\𝔥}$.

The command QuotientAlgebra is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form QuotientAlgebra(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-QuotientAlgebra(...).

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{LieAlgebras}\right)\:$

$\mathrm{L1}≔\mathrm{\_DG}\left(\left[\left[''LieAlgebra''\,\mathrm{Alg1}\,\left[5\right]\right]\,\left[\left[\left[1\,5\,1\right]\,2\right]\,\left[\left[2\,3\,1\right]\,1\right]\,\left[\left[2\,5\,2\right]\,1\right]\,\left[\left[2\,5\,3\right]\,1\right]\,\left[\left[3\,5\,3\right]\,1\right]\,\left[\left[4\,5\,4\right]\,2\right]\right]\right]\right)$

$\mathrm{L1}≔\left[\left[\mathrm{e1}\,\mathrm{e5}\right]\=2\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e5}\right]\=\mathrm{e2}\+\mathrm{e3}\,\left[\mathrm{e3}\,\mathrm{e5}\right]\=\mathrm{e3}\,\left[\mathrm{e4}\,\mathrm{e5}\right]\=2\mathrm{e4}\right]$

$\mathrm{DGsetup}\left(\mathrm{L1}\right)\:$

$\mathrm{h1}≔\left[\mathrm{e1}\,\mathrm{e3}\right]\:$$\mathrm{m1}≔\left[\mathrm{e2}\,\mathrm{e4}\,\mathrm{e5}\right]\:$

$\mathrm{Query}\left(\mathrm{h1}\,''Ideal''\right)$

$\mathrm{true}$

$\mathrm{L2}≔\mathrm{QuotientAlgebra}\left(\mathrm{h1}\,\mathrm{m1}\,\mathrm{Alg2}\right)$

$\mathrm{L2}≔\left[\left[\mathrm{e1}\,\mathrm{e3}\right]\=\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e3}\right]\=2\mathrm{e2}\right]$

$\mathrm{DGsetup}\left(\mathrm{L2}\,\left[f\right]\,\left[\mathrm{\beta }\right]\right)\:$

$\mathrm{L2},A≔\mathrm{QuotientAlgebra}\left(\mathrm{h1}\,\mathrm{m1}\,\mathrm{Alg2}\,''Matrix''\right)$

$\mathrm{L2}\,A:=\left[\left[\left[\left[\mathrm{e1}\,\mathrm{e3}\right]\right]\=\mathrm{e1}\,\left[\left[\mathrm{e2}\,\mathrm{e3}\right]\right]\=2\mathrm{e2}\right]\right]\,\left[\begin{array}{rrrrr}0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right]$

$\mathrm{\Pi }≔\mathrm{Transformation}\left(\mathrm{Alg1}\,\mathrm{Alg2}\,A\right)$

$\mathrm{\Π}≔\left[\left[\mathrm{e1}\,0\mathrm{f1}\right]\,\left[\mathrm{e2}\,\mathrm{f1}\right]\,\left[\mathrm{e3}\,0\mathrm{f1}\right]\,\left[\mathrm{e4}\,\mathrm{f2}\right]\,\left[\mathrm{e5}\,\mathrm{f3}\right]\right]$

$\mathrm{Query}\left(\mathrm{Alg1}\,\mathrm{Alg2}\,\mathrm{\Pi }\,''Homomorphism''\right)$

$\mathrm{true}$

$\mathrm{ApplyHomomorphism}\left(\mathrm{\Pi }\,\mathrm{e1}+2\mathrm{e2}-3\mathrm{e3}+4\mathrm{e4}-5\mathrm{e5}\right)$

$2\mathrm{f1}\+4\mathrm{f2}-5\mathrm{f3}$

$\mathrm{HomomorphismSubalgebras}\left(\mathrm{\Pi }\,''Kernel''\right)$

$\left[\mathrm{e3}\,\mathrm{e1}\right]$

$\mathrm{HomomorphismSubalgebras}\left(\mathrm{\Pi }\,''Image''\right)$

$\left[\mathrm{f1}\,\mathrm{f2}\,\mathrm{f3}\right]$