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LieAlgebras[GeneralizedCenter] - find the generalized center of an ideal

Calling Sequences

     GeneralizedCenter(S1, S2)

Parameters

     S1     - a list of vectors defining a basis for an ideal𝔥 in a Lie algebra 𝔤 

     S2     - (optional) list of vectors defining a basis for a subalgebra k with 𝔥 k 𝔤 .

 

.Description

Examples

DifferentialGeometry, LieAlgebras, Center

.Description

• 

Let 𝔤 be a Lie algebra, k a subalgebra of 𝔤, and 𝔥 an ideal with 𝔥 k 𝔤 . Then the generalized center 𝔠 of h with respect to k is the ideal 𝔠 ={x k  | x, y  h  for all y  k }. In particular, the generalized center of h in 𝔤 is the inverse image of the center of the quotient algebra 𝔤/𝔥 with respect to the canonical projection map π: 𝔤  𝔤/𝔥.

• 

A list of vectors defining a basis for the generalized center of 𝔥 in k is returned. If the optional argument S2 is omitted, then the default is k = 𝔤. If the generalized center of 𝔥 in k is trivial, then an empty list is returned.

• 

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

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

First initialize a Lie algebra.

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

DGsetupL1:

 

Calculate the generalized center of [e1, e2] in the Lie algebra Alg1.

Alg1 > 

S1e1,e2:

Alg1 > 

GeneralizedCentere1,e2

e3,e2,e1

(2.1)

 

Calculate the generalized center of [e1, e4] in [e1, e2, e4, e5].

Alg1 > 

S2e1,e4:S3e1,e2,e4,e5:

Alg1 > 

GeneralizedCenterS2,S3

e5,e1,e2,e4

(2.2)

DifferentialGeometry, LieAlgebras, Center