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
d1 - a sequence of positive integers whose sum equals the dimension of the Lie algebra
Description
Examples
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 and false otherwise
Query(B, [d1, d2, ... ], "DirectSumDecomposition") returns true if the first vectors in , the second vectors in , ... 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(...).
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, spanand span{ . We verify this using Query.
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.
See Also
DifferentialGeometry
LieAlgebras
Decompose
Query
MultiplicationTable
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