Radical - Maple Help
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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 is the largest solvable ideal contained in . The radical of can be calculated as the orthogonal complement of the derived algebra  of with respect to the Killing form , that is, rad = for all . 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  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(is returned. If rad( 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

 

Example 1.

First we initialize a Lie algebra.

(2.1)

 

We calculate the radical of Alg1 to be the 4-dimensional ideal with basis and check that the result is indeed a solvable ideal.

Alg1 > 

(2.2)
Alg1 > 

(2.3)
Alg1 > 

(2.4)

 

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

Alg1 > 

(2.5)
Alg1 > 

(2.6)
Alg1 > 

(2.7)

See Also

DifferentialGeometry

LieAlgebras

LeviDecomposition

Nilradical

Query[Ideal]

Query[Solvable]

 


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