DirectSumDecomposition - Maple Help
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Query[DirectSumDecomposition] - check if a list of subalgebras defines a direct sum decomposition of a Lie algebra

Calling Sequences

Query([S1, S2, ...], "DirectSumDecomposition")

Query(B, [d1, d2, ...], "DirectSumDecomposition")

Parameters

S1        - a list of independent vectors defining a subalgebra of a Lie algebra

B         - a list of vectors defining a basis for $\mathrm{𝔤}$

d1        - a sequence of positive integers whose sum equals the dimension of the Lie algebra $\mathrm{𝔤}$

Description

 • A collection of subalgebras ... of a Lie algebra define a direct sum decomposition of if   (vector space direct sum) and  for .
 • Query([S1, S2, ... ], "DirectSumDecomposition") returns true if the subspaces   define a direct sum decomposition of the Lie algebra $\mathrm{𝔤}$and false otherwise
 • Query(B, [d1, d2, ... ], "DirectSumDecomposition") returns true if the first ${{d}_{1}}_{}$vectors in $B$, the second vectors in $B$, ... define a direct sum decomposition of and false otherwise.
 • The command Query is part of the DifferentialGeometry:-LieAlgebras package.  It can be used in the form Query(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Query(...).

Examples

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

Example 1.

First initialize a Lie algebra and display the Lie bracket multiplication table.  One can see from the multiplication table that this Lie algebra is a direct sum of the subalgebras span, ${S}_{2}=$span${{e}_{4}$and span{ ${e}_{6}}$. We verify this using Query.

 > $\mathrm{L1}≔\mathrm{_DG}\left(\left[\left["LieAlgebra",\mathrm{Alg1},\left[6\right]\right],\left[\left[\left[1,3,1\right],1\right],\left[\left[2,3,2\right],1\right],\left[\left[4,5,4\right],1\right]\right]\right]\right):$
 > $\mathrm{DGsetup}\left(\mathrm{L1}\right):$
 Alg1 > $\mathrm{MultiplicationTable}\left("LieBracket"\right)$
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e4}}\right]$ (2.1)
 Alg1 > $\mathrm{S1}≔\left[\mathrm{e1},\mathrm{e2},\mathrm{e3}\right]:$$\mathrm{S2}≔\left[\mathrm{e4},\mathrm{e5}\right]:$$\mathrm{S3}≔\left[\mathrm{e6}\right]:$
 Alg1 > $\mathrm{Query}\left(\left[\mathrm{S1},\mathrm{S2},\mathrm{S3}\right],"DirectSumDecomposition"\right)$
 ${\mathrm{true}}$ (2.2)
 Alg1 > $\mathrm{Query}\left(\left[\mathrm{S1},\mathrm{S2}\right],"DirectSumDecomposition"\right)$
 ${\mathrm{false}}$ (2.3)

Define to be a basis for the Lie algebra which is adapted to the direct sum decomposition. Use the second calling sequence to check for a direct sum decomposition.

 Alg1 > $B≔\left[\mathrm{e1},\mathrm{e2},\mathrm{e3},\mathrm{e4},\mathrm{e5},\mathrm{e6}\right]:$
 Alg1 > $\mathrm{Query}\left(B,\left[3,3\right],"DirectSumDecomposition"\right)$
 ${\mathrm{true}}$ (2.4)
 Alg1 > $\mathrm{Query}\left(B,\left[2,2,2\right],"DirectSumDecomposition"\right)$
 ${\mathrm{false}}$ (2.5)