Cartan Matrix

A Cartan matrix is a square integer matrix whose elements satisfy the following conditions.

1. is an integer, one of .

2. the diagonal entries are all 2.

3. off of the diagonal.

4. iff .

5. There exists a diagonal matrix such that gives a symmetric and positive definite quadratic form.

A Cartan matrix can be associated to a semisimple Lie algebra . It is a square matrix, where is the Lie algebra rank of . The Lie algebra simple roots are the basis vectors, and is determined by their inner product, using the Killing form.

(1)

In fact, it is more a table of values than a matrix. By reordering the basis vectors, one gets another Cartan matrix, but it is considered equivalent to the original Cartan matrix.

The Lie algebra can be reconstructed, up to isomorphism, by the generators which satisfy the Chevalley-Serre relations. In fact,

(2)

where are the Lie subalgebras generated by the generators of the same letter.

For example,

(3)

is a Cartan matrix. The Lie algebra has six generators . They satisfy the following relations.

1. .

2. and while .

3. .

4. .

5. and .

6. and .

From these relations, it is not hard to see that with the standard Lie algebra representation

			(4)
			(5)
			(6)
			(7)
			(8)
			(9)
			(10)
			(11)

In addition, the Weyl group can be constructed directly from the Cartan matrix, where its rows determine the reflections against the simple roots.

Generalized Cartan matrices determine Kac-Moody algebras, which extend the finite-dimensional theory (Kac 1990).