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

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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

Description

•	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.

•	MinimalSubalgebra(M) calculates the smallest matrix algebra containing the matrices in the list M.

•

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(...).

Examples

Example 1.

First we initialize a Lie algebra and display the multiplication table.

(2.1)

Find the minimal subalgebra containing [e1, e3].

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(2.2)

Find the minimal subalgebra containing [e2, e3].

Alg1 >

(2.3)

Alg1 >

(2.4)

Alg1 >

(2.5)

Find the minimal subalgebra containing [e2, e5].

Alg1 >

(2.6)

Example 2.

The command MinimalSubalgebra also works with matrices.

Alg1 >

M := [Matrix(3, 3, {(1, 1) = 1, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 1, (2, 3) = 0, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 0, (2, 2) = 1, (2, 3) = 1, (3, 1) = 0, (3, 2) = 1, (3, 3) = 0})]

(2.7)

Alg1 >

(2.8)

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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