Query[SymmetricPair] - check if a subalgebra, subspace pair defines a symmetric pair in a Lie algebra
Calling Sequences
Query(M, S, "SymmetricPair")
Query(M, S, parm, "SymmetricPair")
Parameters
M - a list of independent vectors which defines a reductive complement to S in g
S - a list of independent vectors which form a subalgebra in a Lie algebra g
parm - (optional) a set of parameters appearing in the list of vectors M
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Description
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The subspace spanned by M defines a symmetric complement to the subalgebra spanned by S if M is a reductive complement to S and [x, y] in span(S) for all x in M and all y in M.
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If S, M are a symmetric pair then they are always a naturally reductive pair for any inner product B on M.
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Query(M, S, "SymmetricPair") returns true if the subspace M defines a symmetric complement to the subalgebra S, and false otherwise.
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Query(M, S, parm, "SymmetricPair") returns a sequence TF, Eq, Soln, symmetricList. Here TF is true if Maple finds parameter values for which S is a symmetric complement and false otherwise; Eq is the set of equations (with the variables parm as unknowns) which must be satisfied for S to be a symmetric complement; Soln is the list of solutions to the equations Eq; and symmetricList is the list of symmetric complements obtained from the parameter values given by the different solutions in Soln.
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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 initialize a Lie algebra and display the Lie bracket multiplication table.
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We can check that the subspace M1 = [e2, e3] defines a symmetric complement for the subalgebra S1 = [e1, e4].
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In fact, we can show that M1 = [e2, e3] is the only symmetric complement to S1 by constructing the general complement M2 = [e2 + a1*e1 + a2*e4, e3 + a3*e1 + a4*e4].
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SOLN shows that all the parameters must be zero in order for S2, M2 to define a symmetric pair.
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Next we show that the subalgebra S3 = [e4] does not admit a symmetric complement at all.
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| (2.5) |
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| (2.6) |
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