check if a Lie algebra is decomposable as a direct sum of Lie algebras over the real numbers - Maple Programming Help

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Query[Indecomposable] - check if a Lie algebra is decomposable as a direct sum of Lie algebras over the real numbers

Query[AbsolutelyIndecomposable] - check if a Lie algebra is decomposable as a direct sum of Lie algebras over the complex numbers

Calling Sequences

     Query(Alg, "Indecomposable")

     Query(Alg, "AbsolutelyIndecomposable")

Parameters

     Alg     - (optional) the name of an initialized Lie algebra or a Lie algebra data structure

 

Description

Examples

Description

• 

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

• 

Query(Alg, "Indecomposable") returns false if the Lie algebra Alg is decomposable as a direct sum of Lie algebras over the real numbers, otherwise true is returned.

• 

Query(Alg, "AbsolutelyIndecomposable") returns false if the Lie algebra Alg is decomposable as a direct sum of Lie algebras over the complex numbers, otherwise true is returned.

• 

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.

In this example we illustrate the fact that the result of Inquiry("Indecomposable") does not depend upon the choice of basis for the Lie algebra. First we initialize a Lie algebra.

L1_DGLieAlgebra,Alg1,4,1,2,1,1,3,4,3,1

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

(2.1)

DGsetupL1:

 

Now we make a change of basis in the Lie algebra.  In this basis it is not possible to see that the Lie algebra is decomposable by examining the multiplication table.

Alg1 > 

L2LieAlgebraDatae1+e4,e2e3,e2+e4,e1,Alg2

L2:=e1,e2=e1e2+e3+2e4,e1,e3=e4,e2,e3=e1+e2e3e4,e2,e4=e4,e3,e4=e4

(2.2)
Alg1 > 

DGsetupL2:

 

Both Alg1 and Alg2 are seen to be decomposable.

Alg2 > 

QueryAlg1,Indecomposable

false

(2.3)
Alg1 > 

QueryAlg2,Indecomposable

false

(2.4)

 

Example 2

Here is the simplest example of a solvable Lie algebra which is absolutely decomposable but not decomposable. First we initialize the Lie algebra and display the multiplication table.

Alg2 > 

L_DGLieAlgebra,Alg3,4,1,3,1,1,2,4,1,1,1,4,2,1,2,3,2,1

L:=e1,e3=e1,e1,e4=e2,e2,e3=e2,e2,e4=e1

(2.5)
Alg2 > 

DGsetupL:

 

 

The algebra is indecomposable over the real numbers.

Alg3 > 

QueryL,Indecomposable

true

(2.6)

 

The algebra is decomposable over the complex numbers.

Alg3 > 

QueryL,AbsolutelyIndecomposable

false

(2.7)

 

The explicit decomposition of this Lie algebra is given in the help page for the command Decompose.

See Also

DifferentialGeometry

LieAlgebras

Decompose

LieAlgebraData

Query