Query[Indecomposable] - check if a Lie algebra is decomposable as a direct sum of Lie algebras over the real numbers
Query[AbsolutelyIndecomposable] - check if a Lie algebra is decomposable as a direct sum of Lie algebras over the complex numbers
Calling Sequences
Query(Alg, "Indecomposable")
Query(Alg, "AbsolutelyIndecomposable")
Parameters
Alg - (optional) the name of an initialized Lie algebra or a Lie algebra data structure
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Description
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A collection of subalgebras S1, S2, ... of a Lie algebra g defines a direct sum decomposition of g if g = S1 + S2 + ... (vector space direct sum) and [Si, Sj] = 0 for i <> j.
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Query(Alg, "Indecomposable") returns false if the Lie algebra Alg is decomposable as a direct sum of Lie algebras over the real numbers, otherwise true is returned.
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Query(Alg, "AbsolutelyIndecomposable") returns false if the Lie algebra Alg is decomposable as a direct sum of Lie algebras over the complex numbers, otherwise true is returned.
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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.
In this example we illustrate the fact that the result of Inquiry("Indecomposable") does not depend upon the choice of basis for the Lie algebra.
First we initialize a Lie algebra.
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Now we make a change of basis in the Lie algebra. In this basis it is not possible to see that the Lie algebra is decomposable by examining the multiplication table.
Alg1 >
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Alg1 >
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Both Alg1 and Alg2 are seen to be decomposable.
Alg2 >
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Alg1 >
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Example 2
Here is the simplest example of a solvable Lie algebra which is absolutely decomposable but not decomposable.
First we initialize the Lie algebra and display the multiplication table.
Alg2 >
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Alg2 >
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The algebra is indecomposable over the real numbers.
Alg3 >
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The algebra is decomposable over the complex numbers.
Alg3 >
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The explicit decomposition of this Lie algebra is given in the help page for the command Decompose.
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Download Help Document
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