LieAlgebras[Radical] - find the radical of a Lie algebra
Calling Sequences
Radical(LieAlgName)
Parameters
LieAlgName - (optional) name or string, the name of a Lie algebra
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Description
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The radical Radical(g) of a Lie algebra g is the largest solvable ideal contained in g.
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Radical(LieAlgName) calculates the radical of the Lie algebra g defined by LieAlgName. If no argument is given, then the radical of the current Lie algebra is found.
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A list of vectors defining a basis for the radical of g is returned. If the radical of g is trivial, then an empty list is returned.
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The radical of any Lie algebra g can be calculated as the orthogonal complement of the derived algebra of g with respect to the Killing form. See, for example, Fulton and Harris "Representation Theory", Graduate Texts in Mathematics 129, Springer 1991, Proposition C.22 page 484.
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The command Radical is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form Radical(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Radical(...).
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Examples
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Example 1.
First we initialize a Lie algebra.
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We calculate the radical of Alg1 to be the 4 dimensional ideal [e4, e5, e6, e7] and check that the result is indeed a solvable ideal.
Alg1 >
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Alg1 >
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Alg1 >
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We remark that A = [e1, e4, e5, e6, e7] is a solvable subalgebra but it is not an ideal.
Alg1 >
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Alg1 >
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Alg1 >
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