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 - Maple Help

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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 n ×n  matrices

Description

• 

 Let 𝔤 be a Lie algebra and let S1 𝔤 and S2 g be two subspaces (not necessarily subalgebras). Then S1, S2 denotes the span of all vectors of the form x, y with x S1 and y S1. If  S1 = span {x1, x2, ... , xp} and S2 = span {y1, y2, ... , yq} , then

S1, S2 =  span{ xi,  yj | i = 1, 2, ... , p and  j = 1, 2, ... , q}.

Likewise, if M1 and M2 are two subspaces of gln (the Lie algebra of all n ×n matrices), then M1, M2 denotes the span of all matrices of form a, b = ab  ba, with a M1 and b M2.

• 

The first calling sequence BracketOfSubspaces(S1, S2) calculates the subspace S1, S2. A list of linearly independent vectors defining a basis for S1, S2 is returned. If S1, S2 = 0 (that is, all the vectors in S1 commute with all the vectors in S2 ), then an empty list is returned.

• 

The second calling sequence BracketOfSubspaces(M1, M2) calculates the subspace M1, M2. A list of linearly independent vectors defining a basis for M1, M2 is returned. If M1, M2 = 0 (that is, all the matrices in M1 in commute with all the matrices in M2), 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

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

First we initialize a Lie algebra.

L1:=_DGLieAlgebra,Alg1,4,2,4,1,1,3,4,3,1

L1:=e2,e4=e1,e3,e4=e3

(2.1)

DGsetupL1:

 

We bracket the subspaces S1 = span e1, e2 and S2 = span {e3, e4}.

Alg1 > 

S1:=e1,e2:S2:=e3,e4:

Alg1 > 

BracketOfSubspacesS1,S2

e1

(2.2)

 

We bracket the subspace S3 = span{e1,e2, e3} with itself.

Alg1 > 

S3:=e1,e2,e3:

Alg1 > 

BracketOfSubspacesS3,S3

(2.3)

 

Example 2.

The command also works with lists of matrices.

M1:=Matrix0,1,1,0,0,0,0,0,0,Matrix0,1,0,0,0,0,1,0,0

M1:=011000000,010000100

(2.4)

M2:=Matrix0,0,1,0,0,0,0,0,0,Matrix0,0,0,0,0,1,0,0,0,Matrix0,0,0,0,0,0,0,0,1

M2:=001000000,000001000,000000001

(2.5)

BracketOfSubspacesM1,M2

001000000,100000001,001100000,000000100

(2.6)

See Also

DifferentialGeometry, LieAlgebras, Series


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