calculate the span of the Lie bracket of two lists of vectors in a Lie algebra, calculate the span of the matrix commutator of two lists of matrices

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LieAlgebras[BracketOfSubspaces] - calculate the span of the Lie bracket of two lists of vectors in a Lie algebra, calculate the span of the matrix commutator of two lists of matrices

Calling Sequences

BracketOfSubspaces(S1, S2)

BracketOfSubspaces(M1, M2)

Parameters

S1, S2 - two lists of vectors whose spans determine subspaces of a Lie algebra g

M1, M2 - two lists of square matrices

Description

•	The first calling sequence BracketOfSubspaces(S1, S2) calculates the span of the set of vectors [x, y] for all x in S1 and all y in S2.

•	The second calling sequence BracketOfSubspaces(M1, M2) calculates the span of all matrices [a, b] = a.b - b.a for all a in S1 and all b in S2.

•	A list of linearly independent vectors defining a basis for [S1, S2] or [M1, M2] is returned. If [S1, S2] is trivial (that is, all the vectors in S1 commute with all the vectors in S2 then an empty list is returned.

•

The command BracketOfSubspaces is part of the DifferentialGeometry:-LieAlgebras package. It can be used in the form BracketOfSubspaces(...) only after executing the commands with(DifferentialGeometry) and with(LieAlgebras), but can always be used by executing DifferentialGeometry:-LieAlgebras:-BracketOfSubspaces(...).

Examples

Example 1.

First we initialize a Lie algebra.

(2.1)

We bracket the subspaces S1 = [e1, e2] and S2 = [e3, e4].

Alg1 >

(2.2)

We bracket the subspace S3 = [e1, e2, e3] with itself.

Alg1 >

(2.3)

Example 2.

The command also works with lists of matrices.

M1 := [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 1, (3, 2) = 0, (3, 3) = 0})]

(2.4)

M2 := [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 1})]