check if a list of subalgebras defines a direct sum decomposition of a Lie algebra - Maple Programming Help

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Query[DirectSumDecomposition] - check if a list of subalgebras defines a direct sum decomposition of a Lie algebra

Calling Sequences

     Query([S1, S2, ...], "DirectSumDecomposition")

     Query(B, [d1, d2, ...], "DirectSumDecomposition")

Parameters

     S1        - a list of independent vectors defining a subalgebra of a Lie algebra 𝔤   

     B         - a list of vectors defining a basis for 𝔤

     d1        - a sequence of positive integers whose sum equals the dimension of the Lie algebra 𝔤

 

Description

Examples

Description

• 

A collection of subalgebras S1, S2 ... of a Lie algebra 𝔤 define a direct sum decomposition of 𝔤  if  𝔤 = S1S2  (vector space direct sum) and  Si, Sj =0 for i j.

• 

Query([S1, S2, ... ], "DirectSumDecomposition") returns true if the subspaces  S1, S2, ...  define a direct sum decomposition of the Lie algebra 𝔤 and false otherwise

• 

Query(B, [d1, d2, ... ], "DirectSumDecomposition") returns true if the first d1vectors in B, the second d2 vectors in B, ... define a direct sum decomposition of 𝔤 and false otherwise.

• 

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

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

First initialize a Lie algebra and display the Lie bracket multiplication table.  One can see from the multiplication table that this Lie algebra is a direct sum of the subalgebras S1= spane1, e2, e3, S2=span{e4,e5} and S3 =span{ e6}. We verify this using Query.

L1_DGLieAlgebra,Alg1,6,1,3,1,1,2,3,2,1,4,5,4,1:

DGsetupL1:

Alg1 > 

MultiplicationTableLieBracket

e1,e3=e1,e2,e3=e2,e4,e5=e4

(2.1)
Alg1 > 

S1e1,e2,e3:S2e4,e5:S3e6:

Alg1 > 

QueryS1,S2,S3,DirectSumDecomposition

true

(2.2)
Alg1 > 

QueryS1,S2,DirectSumDecomposition

false

(2.3)

 

Define B to be a basis for the Lie algebra which is adapted to the direct sum decomposition. Use the second calling sequence to check for a direct sum decomposition.

Alg1 > 

Be1,e2,e3,e4,e5,e6:

Alg1 > 

QueryB,3,3,DirectSumDecomposition

true

(2.4)
Alg1 > 

QueryB,2,2,2,DirectSumDecomposition

false

(2.5)

See Also

DifferentialGeometry

LieAlgebras

Decompose

Query

MultiplicationTable

 


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