DifferentialGeometry/LieAlgebras/Query/CartanSubalgebra

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Query[CartanSubalgebra] - check if a list of vectors defines a Cartan subalgebra

Calling Sequences

Query()

Parameters

A - a list of vectors, defining a subspace of a Lie algebra

options - one or more of the keyword arguments rank = n (where is a positive integer), algebratype = "Semisimple" or algebratype = "Simple"

Description

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

•	If the Lie algebra is semi-simple and the rank of the Lie algebra is then any Cartan subalgebra is of dimension and is Abelian. This simplifies checking if a given subspace of vectors is a Cartan subalgebra ( the nilpotency of h need not be verified).

Examples

Example 1.

We test if certain subalgebras of are Cartan subalgebras. First define the standard matrix representation for as the space of trace-free matrices.

A := map(convert, [[[1, 0, 0], [0, -1, 0], [0, 0, 0]], [[0, 0, 0], [0, 1, 0], [0, 0, -1]], [[0, 1, 0], [0, 0, 0], [0, 0, 0]], [[0, 0, 1], [0, 0, 0], [0, 0, 0]], [[0, 0, 0], [1, 0, 0], [0, 0, 0]], [[0, 0, 0], [0, 0, 1], [0, 0, 0]], [[0, 0, 0], [0, 0, 0], [1, 0, 0]], [[0, 0, 0], [0, 0, 0], [0, 1, 0]]], Matrix)

(2.1)

Calculate the structure equations for these matrices and initialize the resulting Lie algebra.

(2.2)

(2.3)

Let's check that is semi-simple.

sl3 >

(2.4)

Test to see if a list of vectors defines a Cartan subalgebra.

sl3 >

(2.5)

sl3 >

(2.6)

Since has 2 elements, this implies that the rank of is 2. We can use this information to simplify checking that other subalgebras are Cartan subalgebras

sl3 >

(2.7)

sl3 >

(2.8)

Here is a 2-dimensional Abelian subalgebra which is not self-normalizing and therefore not a Cartan subalgebra.

sl3 >

(2.9)

sl3 >

(2.10)

sl3 >

(2.11)

Example 2.

The notion of a Cartan subalgebra is not restricted to semi-simple Lie algebras. We define a solvable Lie algebra and test to see if some subalgebras are Cartan subalgebras.

sl3 >