LieAlgebras[Query] - check various properties of a Lie algebra, subalgebra, or transformation
Calling Sequence
Query(arg1, arg2, ..., keyword)
Parameters
arg1, - (optional) other arguments
keyword - keyword string
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Description
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The Query function can be used in one of two ways to check various properties of a Lie algebra, subalgebra, or transformation. In the first way the function simply returns true if the property defined by the keyword holds and false otherwise. In the second way, a set of parameters is specified and the function returns the following sequence - TF, Eq, Soln, A. Here TF is true if there is a choice of the parameters which makes the keyword property true; Eq is the list of equations which the parameters must satisfy for the property defined by the keyword to be be true; Soln is the list of all solutions to the equations as found by the Maple solve command; and A is the list of all algebraic structures defined by the previously listed solutions.
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The argument keyword must be one of the following, entered as a string (in quotes " "):
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Abelian AbsolutelyIndecomposable Derivation
DirectSumDecomposition Filtration Gradation
Homomorphism Ideal Indecomposable
Jacobi Keywords LeviDecomposition
NaturallyReductivePair Nilpotent ReductivePair
Semisimple Solvable Subalgebra
SymmetricPair
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Further information is available under ?Query[keyword], where keyword is from the above list.
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A user can add new functionality to Query with the command Query:-addCheck(keyword, procedure).
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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
Alg2 >
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Alg1 >
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Alg2 >
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Alg1 >
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Example 2
In this example we find all the homomorphisms from Alg1 to Alg2 of the form defined by the Matrix A.
Alg1 >
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Alg1 >
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Example 3
In this example we add functionality to Query.
Recall that a Lie algebra is said to be a two-step nilpotent Lie algebra if the second term in the lower central series vanishes.
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f := proc() local C, k;
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Alg2 >
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Alg1 >
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Alg1 >
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Alg1 >
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Alg2 >
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Alg2 >
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Alg1 >
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Note that "two-step" has now been added to the keywords list for Query.
Alg2 >
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