QuotientAlgebra - Maple Help

LieAlgebras[QuotientAlgebra] - find the structure equations for a quotient algebra of a Lie algebra by an ideal

Calling Sequences

QuotientAlgebra(h, m, Algname, keyword)

Parameters

h       - a list of independent vectors defining an ideal  in a Lie algebra $\mathrm{𝔤}$

m       - a list of independent vectors defining a vector space complement to  in $\mathrm{𝔤}$

Algname - a name or a string, the name assigned to the quotient algebra $\mathrm{𝔤}\mathit{/}\mathrm{𝔥}$

keyword - (optional) the keyword "Matrix"

Description

 • Let be a Lie algebra and $\mathrm{𝔥}$ an ideal in $\mathrm{𝔤}$.  Then elements of the quotient algebra are the cosets , where .  The Lie bracket on $\mathrm{𝔤}\mathit{/}\mathrm{𝔥}$ is defined by . If vectors  form a basis for a complement to $\mathrm{𝔥}$, then the cosets  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 $\mathrm{𝔤}\mathit{/}\mathrm{𝔥}$ defined by .
 • 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(...).

Examples

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

Example 1.

First initialize a Lie algebra and display the multiplication table.

 Alg2 > $\mathrm{L1}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg1},\left[5\right]\right],\left[\left[\left[1,5,1\right],2\right],\left[\left[2,3,1\right],1\right],\left[\left[2,5,2\right],1\right],\left[\left[2,5,3\right],1\right],\left[\left[3,5,3\right],1\right],\left[\left[4,5,4\right],2\right]\right]\right]\right)$
 ${\mathrm{L1}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e2}}{+}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e4}}\right]$ (2.1)
 Alg2 > $\mathrm{DGsetup}\left(\mathrm{L1}\right):$

Check that ${h}_{1}$is an ideal and find the quotient algebra (call it Alg2) using the complementary vectors

 Alg1 > $\mathrm{h1}≔\left[\mathrm{e1},\mathrm{e3}\right]:$$\mathrm{m1}≔\left[\mathrm{e2},\mathrm{e4},\mathrm{e5}\right]:$
 Alg1 > $\mathrm{Query}\left(\mathrm{h1},"Ideal"\right)$
 ${\mathrm{true}}$ (2.2)
 Alg1 > $\mathrm{L2}≔\mathrm{QuotientAlgebra}\left(\mathrm{h1},\mathrm{m1},\mathrm{Alg2}\right)$
 ${\mathrm{L2}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{2}{}{\mathrm{e2}}\right]$ (2.3)

Rerun QuotientAlgebra with the keyword argument "Matrix".

 Alg1 > $\mathrm{DGsetup}\left(\mathrm{L2},\left[f\right],\left[\mathrm{\beta }\right]\right):$
 Alg2 > $\mathrm{L2},A≔\mathrm{QuotientAlgebra}\left(\mathrm{h1},\mathrm{m1},\mathrm{Alg2},"Matrix"\right)$
 ${\mathrm{L2}}{,}{A}{:=}\left[\left[\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]\right]{=}{\mathrm{e1}}{,}\left[\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]\right]{=}{2}{}{\mathrm{e2}}\right]\right]{,}\left[\begin{array}{rrrrr}{0}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {1}& {0}\\ {0}& {0}& {0}& {0}& {1}\end{array}\right]$ (2.4)

We use the DifferentialGeometrycommand Transformation to convert the matrix A into a transformation  from Alg1 to the quotient algebra Alg2.

 Alg1 > $\mathrm{\Pi }≔\mathrm{Transformation}\left(\mathrm{Alg1},\mathrm{Alg2},A\right)$
 ${\mathrm{Π}}{≔}\left[\left[{\mathrm{e1}}{,}{0}{}{\mathrm{f1}}\right]{,}\left[{\mathrm{e2}}{,}{\mathrm{f1}}\right]{,}\left[{\mathrm{e3}}{,}{0}{}{\mathrm{f1}}\right]{,}\left[{\mathrm{e4}}{,}{\mathrm{f2}}\right]{,}\left[{\mathrm{e5}}{,}{\mathrm{f3}}\right]\right]$ (2.5)

We can check that $\mathrm{Π}$ is a Lie algebra homomorphism.

 Alg2 > $\mathrm{Query}\left(\mathrm{Alg1},\mathrm{Alg2},\mathrm{\Pi },"Homomorphism"\right)$
 ${\mathrm{true}}$ (2.6)

We see that sends ${e}_{1}$ to 0, ${e}_{2}$ to ${f}_{1}$ and so on.

 Alg2 > $\mathrm{ApplyHomomorphism}\left(\mathrm{\Pi },\mathrm{e1}+2\mathrm{e2}-3\mathrm{e3}+4\mathrm{e4}-5\mathrm{e5}\right)$
 ${2}{}{\mathrm{f1}}{+}{4}{}{\mathrm{f2}}{-}{5}{}{\mathrm{f3}}$ (2.7)

We can verify that [is a basis for the kernel of $\mathrm{Π}$ and that the image of is spanned by (so that is surjective).

 Alg2 > $\mathrm{HomomorphismSubalgebras}\left(\mathrm{\Pi },"Kernel"\right)$
 $\left[{\mathrm{e3}}{,}{\mathrm{e1}}\right]$ (2.8)
 Alg1 > $\mathrm{HomomorphismSubalgebras}\left(\mathrm{\Pi },"Image"\right)$
 $\left[{\mathrm{f1}}{,}{\mathrm{f2}}{,}{\mathrm{f3}}\right]$ (2.9)