find the center of a Lie algebra or a non-commutative algebra - Maple Help

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LieAlgebras[Center] - find the center of a Lie algebra or a non-commutative algebra

Calling Sequences

     Center(Alg)

Parameters

     Alg     - (optional) the name of an initialized Lie algebra

Description

• 

The center of a Lie algebra 𝔤 is the set of all vectors x such that x, y = 0 for all y  𝔤. The center of general algebra 𝔸 is the set of all vectors x such that xy = yx for all y 𝔸.

• 

Center(Alg) returns a list of vectors whose span is the center of the Lie algebra 𝔤 or general algebra 𝔸 defined by Alg. If no argument is given, then the center of the current algebra is found. If the center is trivial, then an empty list is returned.

• 

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

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

First initialize a Lie algebra.

L1:=_DGLieAlgebra,Alg1,5,2,3,1,1,2,5,3,1,4,5,4,1

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

(2.1)

DGsetupL1:

 

Calculate the center of Alg1.

Alg1 > 

CenterAlg1

e1

(2.2)

 

Example 2.

Alg1 > 

L2:=_DGLieAlgebra,Alg2,3,1,2,1,1,1,3,2,2,2,3,3,1

L2:=e1,e2=e1,e1,e3=2e2,e2,e3=e3

(2.3)
Alg1 > 

DGsetupL2:

Alg2 > 

MultiplicationTableLieBracket

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

(2.4)
Alg2 > 

Center

(2.5)

Example 3.

We calculate the center of the octonions.

Alg2 > 

L3:=AlgebraLibraryDataOctonions,Alg3

L3:=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

(2.6)
Alg2 > 

DGsetupL3

algebra name: Alg3

(2.7)
Alg3 > 

Center

e1

(2.8)

 

See Also

DifferentialGeometry, LieAlgebras, Centralizer


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