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LieAlgebras[DerivedAlgebra] - find the derived algebra of a Lie algebra

Calling Sequences

     DerivedAlgebra(LieAlgName)

     DerivedAlgebra(S)

Parameters

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

     S           - a list of vectors defining a basis for a subalgebra of 𝔤

 

Description

Examples

Description

• 

The derived algebra of a Lie algebra 𝔤 is the span of the set of vectors x, y for all x,y 𝔤. The derived algebra is an ideal in 𝔤.

• 

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

• 

DerivedAlgebra(S) calculates the derived algebra of the Lie subalgebra S (viewed as a Lie algebra in its own right).

• 

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

• 

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

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

First we initialize a Lie algebra.

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

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

(2.1)

DGsetupL1:

 

We calculate the derived algebra of Alg1.

Alg1 > 

DerivedAlgebra

e1,e3

(2.2)

 

We calculate the derived algebra of the subalgebra [e1, e2, e4].

Alg1 > 

DerivedAlgebrae1,e2,e4

e1

(2.3)

See Also

DifferentialGeometry

LieAlgebras

BracketOfSubspaces

Series

 


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