LieAlgebras[Complexify] find the complexification of a Lie algebra
Calling Sequences
Complexify(AlgName1, AlgName2)
Parameters
AlgName1 - name or string, the name of a Lie algebra g
AlgName2 - name or string, the name for the complexification of g
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Description
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The complexification of a real Lie algebra g of dimension n is a real Lie algebra of dimension 2n. If e1, e2, ..., en is a basis for g, then e1, e2, ..., en, Ie1, Ie2, ..., Ien, where I^2 = - 1, is a basis for the complexification of g.
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Complexify(AlgName1, AlgName2) calculates the complexification of the Lie algebra g defined by AlgName1.
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A Lie algebra data structure is returned for the complexified Lie algebra with name AlgName2. A Lie algebra data structure contains the structure constants of a Lie algebra in a standard format used by the LieAlgebras package.
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The command Complexify is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form Complexify(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-Complexify(...).
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Examples
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Example 1.
First we initialize a Lie algebra and then display its multiplication table.
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We complexify Alg1 and call the result Alg2.
Alg1 >
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| (2.1) |
Alg1 >
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We note that the original Lie algebra [e1, e2, e3], as a subalgebra of its complexification, admits a symmetric complement.
Alg2 >
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| (2.2) |
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