create the direct sum of a list of Lie algebras - Maple Programming Help

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LieAlgebras[DirectSum] - create the direct sum of a list of Lie algebras

Calling Sequences

     DirectSum(Summands, AlgName)

Parameters

     Summands    - a list of Lie algebra data structures or names of Lie algebras

     AlgName     - a name or string, the name of the direct sum Lie algebra being created

 

Description

Examples

Description

• 

The direct sum of two Lie algebras 𝔤1 and 𝔤2 is the vector space direct sum 𝔤 = 𝔤1  𝔤2 with Lie bracket

 x1, y1, x1, y2=x1, y1,  x2, y2, where  x1, x2   𝔤1  and  y1, y2   𝔤2 .

• 

DirectSum(Summands, AlgName) creates a Lie algebra data structure for the direct sum of the Lie algebras listed in the first argument. The name given to the direct sum algebra is AlgName. The structure equations for the direct sum are displayed.

• 

A Lie algebra data structure contains the structure constants in a standard format used by the LieAlgebras package.  In the LieAlgebras package, the command DGsetup is used to initialize a Lie algebra -- that is, to define the basis elements for the Lie algebra and its dual and to store the structure constants for the Lie algebra in memory.

• 

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

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

First we define 3 Lie algebra data structures and initialize their Lie algebras.  We display the multiplication tables.

L1_DGLieAlgebra,Alg1,2,1,2,1,1:

DGsetupL1,x,a:

Alg1 > 

L2_DGLieAlgebra,Alg2,3,1,2,1,1,1,3,2,2,2,3,3,1:

Alg1 > 

DGsetupL2,y,b:

Alg2 > 

L3_DGLieAlgebra,Alg3,1,:

Alg2 > 

DGsetupL3,z,c:

Alg3 > 

MultiplicationTableAlg1,LieBracket,MultiplicationTableAlg2,LieBracket,MultiplicationTableAlg3,LieBracket

e1,e2=e1,e1,e2=e1,e1,e3=2e2,e2,e3=e3,

(2.1)

 

Create the direct sum of the Lie algebra data structures L1 and L2.

Alg3 > 

L4DirectSumL1,L2,Alg4

L4:=e1,e2=e1,e3,e4=e3,e3,e5=2e4,e4,e5=e5

(2.2)
Alg3 > 

DGsetupL4:

Alg4 > 

MultiplicationTableAlg4,LieTable

| e1e2e3e4e5------------------------e1| 0e1000e2| e10000e3| 000e32e4e4| 00e30e5e5| 002e4e50

(2.3)

 

Create the direct sum of the Lie algebras Alg1, Alg2 and the Lie algebra data structure L3.

Alg4 > 

L5DirectSumAlg1,Alg2,L3,Alg5

L5:=e1,e2=e1,e3,e4=e3,e3,e5=2e4,e4,e5=e5

(2.4)
Alg4   > 

DGsetupL5:

Alg2 > 

MultiplicationTableAlg5,LieTable

| e1e2e3e4e5e6----------------------------e1| 0e10000e2| e100000e3| 000e32e40e4| 00e30e50e5| 002e4e500e6| 000000

(2.5)

See Also

DifferentialGeometry

LieAlgebras

Decompose

Query[DirectSumDecomposition]

 


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