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Query[Semisimple] - check if a Lie algebra is semisimple

Calling Sequences

     Query(Alg, "Semisimple")

     Query(S, "Semisimple")

Parameters

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

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

 

Description

Examples

Description

• 

There are various equivalent definitions for a semisimple Lie algebra. One definition, called Cartan's criterion, states that a Lie algebra is semisimple if its Killing form is non-degenerate. This is the test applied by Query.

• 

Query(Alg, "Semisimple") returns true if Alg is a semisimple Lie algebra and false otherwise.  If the algebra is unspecified, then Query is applied to the current algebra.

• 

Query(S, "Semisimple") returns true if the subalgebra S is a semisimple Lie algebra and false otherwise.

• 

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

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.

First initialize three Lie algebras.

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

L1:=e1,e3=e1,e2,e3=e1+e2

(2.1)

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

(2.2)

L3_DGLieAlgebra,Alg3,5,1,3,2,1,1,4,1,1,2,4,2,1,2,5,1,1,3,4,3,2,3,5,4,1,4,5,5,2

L3:=e1,e3=e2,e1,e4=e1,e2,e4=e2,e2,e5=e1,e3,e4=2e3,e3,e5=e4,e4,e5=2e5

(2.3)

DGsetupL1,x,a:DGsetupL2,y,b:DGsetupL3,z,c:

 

The first and third algebras are not semisimple while the second algebra is semisimple.

Alg3 > 

QueryAlg1,Semisimple

false

(2.4)
Alg1 > 

QueryAlg2,Semisimple

true

(2.5)
Alg2 > 

QueryAlg3,Semisimple

false

(2.6)

 

The subalgebra S = spanz3,  z4, z5 is a semisimple Lie subalgebra of Alg3.

Alg3 > 

Sz3,z4,z5:

Alg3 > 

QueryS,Semisimple

true

(2.7)

See Also

DifferentialGeometry

LieAlgebras

Killing

Query

 


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