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LieAlgebras[Centralizer] - find the centralizer of a list of vectors

Calling Sequences

     Centralizer(S, h)

Parameters

     S     - a list of vectors in a Lie algebra 𝔤 or a general algebra 𝔸

     h     - (optional) a subalgebra of 𝔤 or 𝔸

 

Description

Examples

Description

• 

The centralizer of a set of vectors S relative to a subalgebra h is the subalgebra of vectors in h which commute with all the vectors in S.

• 

Centralizer(S, h) calculates the centralizer of the list Sin the subalgebra h. If the second argument h is not specified then the centralizer of S in the entire algebra is calculated.

• 

A list of vectors defining a basis for the centralizer of S is returned. If the centralizer of S is trivial, then an empty list is returned.

• 

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

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

First initialize a Lie algebra.

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

L1:=e2,e5=e1,e3,e4=e1,e3,e5=e2

(2.1)

DGsetupL1:

 

Calculate the centralizer of  e3 in the Lie algebra Alg1.

Alg1 > 

Se3

S:=e3

(2.2)
Alg1 > 

CentralizerS

e3,e2,e1

(2.3)

 

Calculate the centralizer of e4, e5 relative to the subalgebras spanned by e1,e2, e4, e5 and  e1,e2, e3.

Alg1 > 

Se4,e5

S:=e4,e5

(2.4)
Alg1 > 

he1,e2,e4,e5

h:=e1,e2,e4,e5

(2.5)
Alg1 > 

CentralizerS,h

e5,e4,e1

(2.6)
Alg1 > 

he1,e2,e3

h:=e1,e2,e3

(2.7)
Alg1 > 

CentralizerS,h

e1

(2.8)

 

Example 2.

Calculate the centralizer of a set of vectors in the algebra 𝕆 of octonions.

Alg1 > 

L2AlgebraLibraryDataOctonions,Oct

L2:=e12=e1,e1.e2=e2,e1.e3=e3,e1.e4=e4,e1.e5=e5,e1.e6=e6,e1.e7=e7,e1.e8=e8,e2.e1=e2,e22=e1,e2.e3=e4,e2.e4=e3,e2.e5=e6,e2.e6=e5,e2.e7=e8,e2.e8=e7,e3.e1=e3,e3.e2=e4,e32=e1,e3.e4=e2,e3.e5=e7,e3.e6=e8,e3.e7=e5,e3.e8=e6,e4.e1=e4,e4.e2=e3,e4.e3=e2,e42=e1,e4.e5=e8,e4.e6=e7,e4.e7=e6,e4.e8=e5,e5.e1=e5,e5.e2=e6,e5.e3=e7,e5.e4=e8,e52=e1,e5.e6=e2,e5.e7=e3,e5.e8=e4,e6.e1=e6,e6.e2=e5,e6.e3=e8,e6.e4=e7,e6.e5=e2,e62=e1,e6.e7=e4,e6.e8=e3,e7.e1=e7,e7.e2=e8,e7.e3=e5,e7.e4=e6,e7.e5=e3,e7.e6=e4,e72=e1,e7.e8=e2,e8.e1=e8,e8.e2=e7,e8.e3=e6,e8.e4=e5,e8.e5=e4,e8.e6=e3,e8.e7=e2,e82=e1

(2.9)
Alg1 > 

DGsetupL2

algebra name: Oct

(2.10)
Alg1 > 

Se3

S:=e3

(2.11)
Alg1 > 

CentralizerS

e1,e3

(2.12)

 

See Also

DifferentialGeometry

LieAlgebras

Center

 


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