Query[Filtration] - check if a list of subspaces defines a decreasing filtration of a Lie algebra
Calling Sequences
Query([f0, f1, ..., fN], "Filtration")
Parameters
f0, f1, - a list of independent vectors defining subspaces of a Lie algebra g
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Description
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A collection of subspaces f0, f1, ... fN of a Lie algebra g defines a decreasing filtration of g if fi in fj for i >= j, [fi, fj] in f(i + j) for i + j <=N and [fi, fj] in fN for i + j > N.
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Query([f0, f1, ... fN], "Filtration") returns true if the subspaces f0, f1, ..., fN define a decreasing filtration of the Lie algebra 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.
First we initialize a Lie algebra.
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Now define a sequence of 4 subspaces.
We check that this sequence of Lie algebras defines a decreasing filtration.
Alg1 >
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Example 2.
Here's an example which does not define a filtration. To see the specific brackets which fail to satisfy the filtration definition, we set the infolevel for Query to 2.
Alg1 >
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Alg1 >
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bracket of subspace 0 and 0 is [2*e1, e2, e2+e3]
bracket of subspace 0 and 1 is [-e1, -e2]
bracket of subspace 0 and 2 is [-e1, -e2]
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This shows that [f0, f2] is not contained in f2.
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