find a basis for a solvable Lie algebra which defines an ascending chain of ideals - Maple Programming Help

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LieAlgebras[AscendingIdealsBasis] - find a basis for a solvable Lie algebra which defines an ascending chain of ideals

Calling Sequences

AscendingIdealsBasis(Alg)

Parameters

Alg        - (optional) Maple name or string, the name of an initialized Lie algebra

Description

 • Every (complex) solvable Lie algebra admits a basis such that the subspace spanform an ideal in span The command AscendingIdealsBasis calculates such a basis. This basis can be quite useful in a situation where the matrix exponentials of the adjoint matrices are needed.

Examples

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

Example 1.

First we initialize a 5-dimensional Lie algebra.

 > $L≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg1},\left[5\right]\right],\left[\left[\left[1,2,1\right],-22\right],\left[\left[1,2,2\right],-11\right],\left[\left[1,2,3\right],21\right],\left[\left[1,2,4\right],1\right],\left[\left[1,2,5\right],11\right],\left[\left[1,3,1\right],3\right],\left[\left[1,3,2\right],2\right],\left[\left[1,3,3\right],-4\right],\left[\left[1,4,1\right],1\right],\left[\left[1,4,5\right],-2\right],\left[\left[1,5,1\right],-12\right],\left[\left[1,5,2\right],-7\right],\left[\left[1,5,3\right],13\right],\left[\left[1,5,4\right],1\right],\left[\left[1,5,5\right],3\right],\left[\left[2,3,1\right],12\right],\left[\left[2,3,2\right],6\right],\left[\left[2,3,3\right],-11\right],\left[\left[2,3,4\right],-1\right],\left[\left[2,3,5\right],-6\right],\left[\left[2,4,1\right],19\right],\left[\left[2,4,2\right],9\right],\left[\left[2,4,3\right],-19\right],\left[\left[2,4,4\right],1\right],\left[\left[2,4,5\right],-11\right],\left[\left[2,5,1\right],16\right],\left[\left[2,5,2\right],8\right],\left[\left[2,5,3\right],-16\right],\left[\left[2,5,5\right],-8\right],\left[\left[3,4,1\right],2\right],\left[\left[3,4,3\right],-1\right],\left[\left[3,4,4\right],1\right],\left[\left[3,4,5\right],-4\right],\left[\left[3,5,1\right],-8\right],\left[\left[3,5,2\right],-5\right],\left[\left[3,5,3\right],9\right],\left[\left[3,5,4\right],1\right],\left[\left[3,5,5\right],1\right],\left[\left[4,5,1\right],-7\right],\left[\left[4,5,2\right],-4\right],\left[\left[4,5,3\right],7\right],\left[\left[4,5,4\right],1\right],\left[\left[4,5,5\right],2\right]\right]\right]\right):$
 > $\mathrm{DGsetup}\left(L\right)$
 ${\mathrm{Lie algebra: Alg1}}$ (2.1)

We can use the command Query/"Solvable" to check that this is a solvable Lie algebra.

 Alg1 > $\mathrm{Query}\left("Solvable"\right)$
 ${\mathrm{true}}$ (2.2)

Now we calculate a basis with the ascending ideals property.

 Alg1 > $B≔\mathrm{AscendingIdealsBasis}\left(\right)$
 ${B}{:=}\left[{\mathrm{e1}}{-}{2}{}{\mathrm{e5}}{,}{\mathrm{e2}}{-}{2}{}{\mathrm{e4}}{+}{3}{}{\mathrm{e5}}{,}{\mathrm{e3}}{-}{\mathrm{e4}}{,}{\mathrm{e1}}{,}{\mathrm{e2}}\right]$ (2.3)

The following two commands check, for example, that  span $B\left[1..3\right]$ is an ideal in span $B\left[1..4\right]$.

 Alg1 > $C≔\mathrm{BracketOfSubspaces}\left({B}_{1..3},{B}_{1..4}\right)$
 ${C}{:=}\left[{-}{24}{}{\mathrm{e1}}{-}{14}{}{\mathrm{e2}}{+}{26}{}{\mathrm{e3}}{+}{2}{}{\mathrm{e4}}{+}{6}{}{\mathrm{e5}}{,}{60}{}{\mathrm{e1}}{+}{32}{}{\mathrm{e2}}{-}{60}{}{\mathrm{e3}}{-}{4}{}{\mathrm{e4}}{-}{24}{}{\mathrm{e5}}{,}{-}{2}{}{\mathrm{e1}}{-}{2}{}{\mathrm{e2}}{+}{4}{}{\mathrm{e3}}{-}{2}{}{\mathrm{e5}}\right]$ (2.4)
 Alg1 > $\mathrm{GetComponents}\left(C,{B}_{1..3},\mathrm{trueorfalse}="on"\right)$
 ${\mathrm{true}}$ (2.5)

The command  Query/"AscendingIdealsBasis" will verify that the basis B has the ascending ideals property.

 Alg1 > $\mathrm{Query}\left(B,"AscendingIdealsBasis"\right)$
 ${\mathrm{true}}$ (2.6)

The ascending ideals property becomes apparent if we re-initialize the Lie algebra using the basis B (using the command LieAlgebraData).

 Alg1 > $\mathrm{L2}≔\mathrm{LieAlgebraData}\left(B,\mathrm{alg2}\right)$
 ${\mathrm{L2}}{:=}\left[\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{24}{}{\mathrm{e1}}{-}{14}{}{\mathrm{e2}}{+}{26}{}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{10}{}{\mathrm{e1}}{+}{5}{}{\mathrm{e2}}{-}{11}{}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{60}{}{\mathrm{e1}}{+}{32}{}{\mathrm{e2}}{-}{60}{}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{-}{10}{}{\mathrm{e1}}{-}{6}{}{\mathrm{e2}}{+}{10}{}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{2}{}{\mathrm{e1}}{-}{2}{}{\mathrm{e2}}{+}{4}{}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{7}{}{\mathrm{e1}}{+}{3}{}{\mathrm{e2}}{-}{8}{}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{22}{}{\mathrm{e1}}{-}{11}{}{\mathrm{e2}}{+}{21}{}{\mathrm{e3}}\right]$ (2.7)
 Alg1 > $\mathrm{DGsetup}\left(\mathrm{L2}\right)$
 ${\mathrm{Lie algebra: alg2}}$ (2.8)
 alg2 > $\mathrm{MultiplicationTable}\left("LieTable"\right)$
 $\left[\begin{array}{ccccccc}{}& {|}& {\mathrm{e1}}& {\mathrm{e2}}& {\mathrm{e3}}& {\mathrm{e4}}& {\mathrm{e5}}\\ {}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}& {\mathrm{----}}\\ {\mathrm{e1}}& {|}& {0}& {0}& {0}& {-}{24}{}{\mathrm{e1}}{-}{14}{}{\mathrm{e2}}{+}{26}{}{\mathrm{e3}}& {10}{}{\mathrm{e1}}{+}{5}{}{\mathrm{e2}}{-}{11}{}{\mathrm{e3}}\\ {\mathrm{e2}}& {|}& {0}& {0}& {0}& {60}{}{\mathrm{e1}}{+}{32}{}{\mathrm{e2}}{-}{60}{}{\mathrm{e3}}& {-}{10}{}{\mathrm{e1}}{-}{6}{}{\mathrm{e2}}{+}{10}{}{\mathrm{e3}}\\ {\mathrm{e3}}& {|}& {0}& {0}& {0}& {-}{2}{}{\mathrm{e1}}{-}{2}{}{\mathrm{e2}}{+}{4}{}{\mathrm{e3}}& {7}{}{\mathrm{e1}}{+}{3}{}{\mathrm{e2}}{-}{8}{}{\mathrm{e3}}\\ {\mathrm{e4}}& {|}& {24}{}{\mathrm{e1}}{+}{14}{}{\mathrm{e2}}{-}{26}{}{\mathrm{e3}}& {-}{60}{}{\mathrm{e1}}{-}{32}{}{\mathrm{e2}}{+}{60}{}{\mathrm{e3}}& {2}{}{\mathrm{e1}}{+}{2}{}{\mathrm{e2}}{-}{4}{}{\mathrm{e3}}& {0}& {-}{22}{}{\mathrm{e1}}{-}{11}{}{\mathrm{e2}}{+}{21}{}{\mathrm{e3}}\\ {\mathrm{e5}}& {|}& {-}{10}{}{\mathrm{e1}}{-}{5}{}{\mathrm{e2}}{+}{11}{}{\mathrm{e3}}& {10}{}{\mathrm{e1}}{+}{6}{}{\mathrm{e2}}{-}{10}{}{\mathrm{e3}}& {-}{7}{}{\mathrm{e1}}{-}{3}{}{\mathrm{e2}}{+}{8}{}{\mathrm{e3}}& {22}{}{\mathrm{e1}}{+}{11}{}{\mathrm{e2}}{-}{21}{}{\mathrm{e3}}& {0}\end{array}\right]$ (2.9)