LieAlgebras[DirectSum] - create the direct sum of a list of Lie algebras
Calling Sequences
DirectSum(Summands, AlgName)
Parameters
Summands - a list of Lie algebra data structures or names of Lie algebras
AlgName - a name or string, the name of the direct sum Lie algebra being created
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Description
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The direct sum of two Lie algebras g1 and g2 is the vector space direct sum g = g1 + g2, with Lie bracket [(x1, x2), (y1, y2)] = [x1, y1] + [x2, y2] where x1, y1 in g1 and x2, y2 in g2. This definition extends in the natural way to the direct sum of several algebras.
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DirectSum(Summands, AlgName) creates a Lie algebra data structure for the direct sum of the Lie algebras listed in the first argument. The name given to the direct sum algebra is AlgName.
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A Lie algebra data structure contains the structure constants in a standard format used by the LieAlgebras package. In the LieAlgebras package, the command DGsetup is used to initialize a Lie algebra -- that is, to define the basis elements for the Lie algebra and its dual and to store the structure constants for the Lie algebra in memory.
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The command DirectSum is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form DirectSum(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-DirectSum(...).
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Examples
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Example 1.
First we define 3 Lie algebra data structures and initialize their Lie algebras. We display the multiplication tables.
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Alg1 >
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Alg1 >
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Alg2 >
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Alg2 >
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Alg3 >
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Create the direct sum of the Lie algebra data structures L1 and L2.
Alg3 >
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Alg3 >
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Alg4 >
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Create the direct sum of the Lie algebras Alg1, Alg2 and the Lie algebra data structure L3.
Alg4 >
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Alg4 >
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Alg2 >
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| (2.5) |
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