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LieAlgebras[Radical] - find the radical of a Lie algebra

Calling Sequences

     Radical(LieAlgName)

Parameters

     LieAlgName - (optional) name or string, the name of a Lie algebra

 

Description

Examples

Description

• 

The radical of a Lie algebra 𝔤 is the largest solvable ideal contained in 𝔤. The radical of 𝔤 can be calculated as the orthogonal complement of the derived algebra 𝔤' of 𝔤 with respect to the Killing form B, that is, rad𝔤 = {x  𝔤 | Bx, y = 0 for all y 𝔤'}. See, for example, Fulton and Harris Representation Theory, Graduate Texts in Mathematics 129, Springer 1991, Proposition C.22 page 484.

• 

Radical(LieAlgName) calculates the radical of the Lie algebra 𝔤 defined by LieAlgName. If no argument is given, then the radical of the current Lie algebra is found.

• 

A list of vectors defining a basis for the rad(𝔤)is returned. If rad(𝔤) is trivial, then an empty list is returned.

• 

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

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

First we initialize a Lie algebra.

L1_DGLieAlgebra,Alg1,7,1,2,2,2,1,3,3,2,2,3,1,1,1,4,4,1,1,5,5,1,2,5,4,1,3,4,5,1,4,5,6,1,4,7,4,1,5,7,5,1,6,7,6,2

L1:=e1,e2=2e2,e1,e3=2e3,e1,e4=e4,e1,e5=e5,e2,e3=e1,e2,e5=e4,e3,e4=e5,e4,e5=e6,e4,e7=e4,e5,e7=e5,e6,e7=2e6

(2.1)

DGsetupL1:

 

We calculate the radical of Alg1 to be the 4-dimensional ideal with basis e4,e5,e6,e7and check that the result is indeed a solvable ideal.

Alg1 > 

radRadical

rad:=e7,e6,e5,e4

(2.2)
Alg1 > 

Queryrad,Solvable

true

(2.3)
Alg1 > 

Queryrad,Ideal

true

(2.4)

 

We remark that the span of the vectors e1,e4,e6,e7is a 4-dimensional solvable subalgebra but it is not an ideal.

Alg1 > 

Ae1,e4,e5,e6,e7

A:=e1,e4,e5,e6,e7

(2.5)
Alg1 > 

QueryA,Solvable

true

(2.6)
Alg1 > 

QueryA,Ideal

false

(2.7)

See Also

DifferentialGeometry

LieAlgebras

LeviDecomposition

Nilradical

Query[Ideal]

Query[Solvable]