LieAlgebras[MinimalSubalgebra] - find the smallest Lie subalgebra containing a given set of vectors from a Lie algebra, find the smallest matrix algebra containing a given set of matrices
Calling Sequences
MinimalSubalgebra(S)
MinimalSubalgebra(M)
Parameters
S - a list of vectors in a Lie algebra
M - a list of square matrices
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Description
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MinimalSubalgebra(S) calculates the smallest Lie subalgebra containing the list of vectors S from a defined Lie algebra g. A list of basis vectors for the subalgebra from the Lie algebra g is returned.
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MinimalSubalgebra(M) calculates the smallest matrix algebra containing the matrices in the list M.
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The command MinimalSubalgebra is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form MinimalSubalgebra(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-MinimalSubalgebra(...).
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Examples
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Example 1.
First we initialize a Lie algebra and display the multiplication table.
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Find the minimal subalgebra containing [e1, e3].
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Alg1 >
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Find the minimal subalgebra containing [e2, e3].
Alg1 >
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Alg1 >
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Alg1 >
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| (2.4) |
Alg1 >
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Find the minimal subalgebra containing [e2, e5].
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Alg1 >
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Example 2.
The command MinimalSubalgebra also works with matrices.
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Alg1 >
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We can use the LieAlgebraData command to verify that the set of matrices N defines a 4-dimensional Lie algebra and to determine the commutator relationships.
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| (2.9) |
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Here e1, e2, e3, e4 denote the four matrices N[1], N[2], N[3], N[4].
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