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LieAlgebras[SubalgebraNormalizer] - find the normalizer of a subalgebra

Calling Sequences

     SubalgebraNormalizer(h, k)

Parameters

     h         - a list of vectors defining a subalgebra h in a Lie algebra 𝔤

     k         - (optional) a list of vectors defining a subalgebra k of 𝔤 containing the subalgebra h

 

Description

Examples

Description

• 

Let 𝔤 be a Lie algebra and let h k 𝔤 be subalgebras.The normalizer n of h in k is the largest subalgebra n of k which contains h as an ideal. The normalizer of h always contains h itself.

• 

SubalgebraNormalizer(h, k) calculates the normalizer of h in the subalgebra k. If the second argument k is not specified, then the default is k =𝔤 and the normalizer of h in 𝔤 is calculated.

• 

A list of vectors defining a basis for the normalizer of h is returned.

• 

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

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

First initialize a Lie algebra and display the Lie bracket multiplication table.

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

Alg1   > 

DGsetupL1:

MultiplicationTableLieBracket

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

(2.1)

 

Calculate the normalizer of S1 = span e3 in S2 =spane1,e3,e4.

Alg1 > 

S1e3:S2e1,e3,e4:

Alg1 > 

SubalgebraNormalizerS1,S2

e3,e1

(2.2)

 

Calculate the normalizer of S3=spane2,e4 in S4=e1, e2,e4,e5.

Alg1 > 

S3e2,e4:S4e1,e2,e4,e5:

Alg1 > 

SubalgebraNormalizerS3,S4

e4,e2,e1

(2.3)

 

Calculate the normalizer of S5=spane1,e2 in the Lie algebra Alg1.

Alg1 > 

S5e1,e2:

Alg1 > 

SubalgebraNormalizerS5

e5,e4,e3,e2,e1

(2.4)

See Also

DifferentialGeometry

LieAlgebras

Centralizer

MultiplicationTable

Query[ideal]

Query[nilpotent]