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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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![L1 := _DG([["LieAlgebra", Alg1, [4]], [[[1, 2, 1], 1], [[3, 4, 3], 1]]])](/support/helpjp/helpview.aspx?si=7162/file07653/math79.png)
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![L1 := [[e1, e2] = e1, [e3, e4] = e3]](/support/helpjp/helpview.aspx?si=7162/file07653/math82.png)
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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.
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Alg1 >
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![L2 := LieAlgebraData([e1+e4, e2-e3, e2+e4, e1], Alg2)](/support/helpjp/helpview.aspx?si=7162/file07653/math94.png)
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![L2 := [[e1, e2] = -e1-e2+e3+2*e4, [e1, e3] = e4, [e2, e3] = e1+e2-e3-e4, [e2, e4] = -e4, [e3, e4] = -e4]](/support/helpjp/helpview.aspx?si=7162/file07653/math97.png)
| (2.2) |
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Alg1 >
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Both Alg1 and Alg2 are seen to be decomposable.
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Alg2 >
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| (2.3) |
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Alg1 >
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| (2.4) |
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.
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Alg2 >
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![L := _DG([["LieAlgebra", Alg3, [4]], [[[1, 3, 1], 1], [[2, 4, 1], 1], [[1, 4, 2], -1], [[2, 3, 2], 1]]])](/support/helpjp/helpview.aspx?si=7162/file07653/math135.png)
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![L := [[e1, e3] = e1, [e1, e4] = -e2, [e2, e3] = e2, [e2, e4] = e1]](/support/helpjp/helpview.aspx?si=7162/file07653/math138.png)
| (2.5) |
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Alg2 >
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The algebra is indecomposable over the real numbers.
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Alg3 >
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| (2.6) |
The algebra is decomposable over the complex numbers.
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Alg3 >
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| (2.7) |
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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