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LieAlgebras[CartanSubalgebra] - find a Cartan subalgebra of a Lie algebra

Calling Sequences

     CartanSubalgebra()

     CartanSubalgebra(alg)

     CartanSubalgebra(N)

Parameters

     alg   - name or string, the name of an initialized Lie algebra

  N     - a list of vectors, defining a nilpotent subalgebra

   

 

Description

Examples

Description

• 

 Let 𝔤 be a Lie algebra. A Cartan subalgebra h is a nilpotent subalgebra whose normalizer in g is itself, that is, nor𝔥 = 𝔥 . If g is a semi-simple Lie algebra, then every Cartan subalgebra h is Abelian and adx (see Adjoint) is a semi-simple linear transformation for every x 𝔥 (that is, adx is diagonalizable over C). Cartan subalgebras are not unique. However, if g is a semi-simple Lie algebra, then any two Cartan subalgebras of g are related by an automorphism of g Let n be a nilpotent subalgebra of g and let F𝔫 ={y 𝔤 | adxry =0} be the generalized null space of n. Then there always exists a Cartan subalgebra 𝔥 F𝔫. If x 𝔤 is a regular element, then the generalized null space of x is a Cartan subalgebra.

• 

The procedure CartanSubalgebra returns a list of vectors whose span is a Cartan subalgebra.

• 

The procedure CartanSubalgebra implements the algorithm for calculating Cartan subalgebras presented in W. A. De Graaf: Lie Algebras: Theory and Algorithms.

Examples

 

withDifferentialGeometry:withLieAlgebras:withLibrary:

 

Example 1

We calculate the Cartan subalgebra for sl3, the 8-dimensional Lie algebra of 3x3 trace-free matrices. The structure equations are obtained using the SimpleLieAlgebraData command.

LSimpleLieAlgebraDatasl(3),sl3,labelformat=gl,labels=E,θ

L:=e1,e3=e3,e1,e4=2e4,e1,e5=e5,e1,e6=e6,e1,e7=2e7,e1,e8=e8,e2,e3=e3,e2,e4=e4,e2,e5=e5,e2,e6=2e6,e2,e7=e7,e2,e8=2e8,e3,e5=e1e2,e3,e6=e4,e3,e7=e8,e4,e5=e6,e4,e7=e1,e4,e8=e3,e5,e8=e7,e6,e7=e5,e6,e8=e2,E11,E22,E12,E13,E21,E23,E31,E32,θ11,θ22,θ12,θ13,θ21,θ23,θ31,θ32

(2.1)

 

Initialized the Lie algebra.

DGsetupL

Lie algebra: sl3

(2.2)

 

Find a Cartan subalgebra.

sl(3) > 

CSACartanSubalgebra

CSA:=E11,E22

(2.3)

 

We can check that this subalgebra is Abelian (and hence nilpotent) and self-normalizing.

sl3 > 

QueryCSA,Abelian

true

(2.4)
sl3 > 

SubalgebraNormalizerCSA

E22,E11

(2.5)

 

These properties can also be checked with the Query command

sl3 > 

QueryCSA,CartanSubalgebra

true

(2.6)

 

For the split real forms of the simple Lie algebras, a Cartan subalgebra can always be found consisting of diagonal matrices in the standard representation.

sl3 > 

StandardRepresentationsl3,CSA

100000001,000010001

(2.7)

 

Example 2

Other Cartan subalgebras for sl4 can be found with the second calling sequence.

sl3 > 

CartanSubalgebraE11+3E31

E11+3E31,E22+32E31

(2.8)

 

Example 3

The Cartan subalgebra of a nilpotent Lie algebra g is g itself. Retrieve the structure equations for a nilpotent Lie algebra from the DifferentialGeometry library.

sl3 > 

LD3RetrieveWinternitz,1,5,5,alg3

LD3:=e2,e5=e1,e3,e4=e1,e3,e5=e2

(2.9)
sl3 > 

DGsetupLD3:

 

Check that the algebra is nilpotent.

alg3 > 

QueryNilpotent

true

(2.10)
alg3 > 

CartanSubalgebra

e1,e2,e3,e4,e5

(2.11)

 

Example 4

We find the Cartan subalgebra for a solvable Lie algebra. Retrieve the structure equations for a solvable Lie algebra from the DifferentialGeometry library.

alg3 > 

LD4RetrieveWinternitz,1,5,34,alg4

LD4:=e1,e4=ae1,e1,e5=e1,e2,e4=e2,e3,e4=e3,e3,e5=e2

(2.12)
alg4 > 

DGsetupLD4

Lie algebra: alg4

(2.13)

 

Check that the algebra is solvable.

alg4 > 

QuerySolvable

true

(2.14)
alg4 > 

CSA4CartanSubalgebra

CSA4:=e4,e5

(2.15)
alg4 > 

QueryCSA4,CartanSubalgebra

true

(2.16)

 

Example 5.

We find the Cartan subalgebra for a Lie algebra with a non-trivial Levi decomposition. Retrieve the structure equations for such a Lie algebra from the DifferentialGeometry library.

alg4 > 

LD5RetrieveTurkowski,1,7,4,alg5

LD5:=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

(2.17)
alg4 > 

DGsetupLD5

Lie algebra: alg5

(2.18)

 

Check that the Levi decomposition is non-trivial.

alg5 > 

LeviDecomposition

e4,e5,e6,e7,e1,e2,e3

(2.19)

 

Calculate the Cartan subalgebra.

alg5 > 

CartanSubalgebraalg5

e1,e7

(2.20)

See Also

DifferentialGeometry

LieAlgebras

CartanMatrix

Query

RootSpaceDecomposition

 


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