find a Cartan subalgebra of a Lie algebra

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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 intialized Lie algebra

N - a list of vectors, defining a nilpotent subalgebra

Description

•	Let be a Lie algebra. A Cartan subalgebra h is nilpotent subalgebra whose normalizer in g is itself, that is, .

•	If g is a semi-simple Lie algebra, then any Cartan subalgebra h is Abelian and (see Adjoint) is a semi-simple linear transformation for every (that is, 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 $F(`&nfr;` ) = {y in `&gfr;` \| ad(x)^(r)(y) =0 }$ be the generalized null space of n. Then there always exists a Cartan subalgebra

•	If is a regular element, then the generalized null space of 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

Example 1

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

(2.1)

Initialized the Lie algebra.

(2.2)

Find a Cartan subalgebra.

sl(3) >

(2.3)

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

sl3 >

(2.4)

sl3 >

(2.5)

These properties can also be checked with the Query command

sl3 >

(2.6)

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

sl3 >

[Matrix(3, 3, {(1, 1) = 1, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = -1}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 0, (2, 1) = 0, (2, 2) = 1, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = -1})]

(2.7)

Example 2

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

sl3 >

(2.8)

Example 3

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

sl3 >

(2.9)

sl3 >

Check that the algebra is nilpotent.

alg3 >

(2.10)

alg3 >

(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 >

(2.12)

alg4 >

(2.13)

Check that the algebra is solvable.

alg4 >

(2.14)

alg4 >

(2.15)

alg4 >

(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 >