Query[LeviDecomposition] - check that a pair of subalgebras define a Levi decomposition of a Lie algebra
Calling Sequences
Query([R, S], "LeviDecomposition")
Parameters
R - a list of independent vectors in a Lie algebra g
S - a list of independent vectors in a Lie algebra g
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Description
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A pair of subalgebras [R, S] in a Lie algebra define a Levi decomposition if R is the radical of g, S is a semisimple subalgebra, and g = R + S (vector space direct sum). Since the radical is an ideal we have [R, R] in R, [R, S] in R, and [S, S] in S. The radical R is unique, the semisimple subalgebra S in a Levi decomposition is not.
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Query([R, S], "LeviDecomposition") returns true if the pair R, S is a Levi decomposition of g and false otherwise.
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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(...).
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Examples
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Example 1.
We initialize three different Lie algebras and print their multiplication tables.
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Alg3 >
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Alg1 is solvable and therefore the radical is the entire algebra.
Alg3 >
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Alg3 >
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| (2.2) |
Alg2 is semisimple and therefore the radical is the zero subalgebra.
Alg1 >
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Alg1 >
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Alg3 has a non-trivial Levi decomposition.
Alg1 >
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Alg1 >
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It is easy to see that in this last example the Levi decomposition is not unique.
First we find the general complement to the radical R3 using the ComplementaryBasis program.
Alg1 >
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| (2.5) |
Next we determine for which values of the parameters {k1, k2, k3, k4, k5, k6} the subspace SS0 is a Lie subalgebra. We find that k1 = 0, k2 = k3, k4 = - k5, k6 = 0.
Alg3 >
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Alg3 >
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| (2.7) |
Alg3 >
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| (2.8) |
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