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Cartan Matrix -- from Wolfram MathWorld

  • ️Weisstein, Eric W.
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A Cartan matrix is a square integer matrix who elements (A_(ij)) satisfy the following conditions.

1. A_(ij) is an integer, one of {-3,-2,-1,0,2}.

2. A_(ii)=2 the diagonal entries are all 2.

3. A_(ij)<=0 off of the diagonal.

4. A_(ij)=0 iff A_(ji)=0.

5. There exists a diagonal matrix D such that DAD^(-1) gives a symmetric and positive definite quadratic form.

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

A_(ij)=2<alpha_i,alpha_j>/<alpha_j,alpha_j>

(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 g can be reconstructed, up to isomorphism, by the 3k generators {e_i,f_i,h_i} which satisfy the Chevalley-Serre relations. In fact,

g=h direct sum e direct sum f

(2)

where h,e,f are the Lie subalgebras generated by the generators of the same letter.

For example,

A=[ 2 -1; -1 2]

(3)

is a Cartan matrix. The Lie algebra g has six generators {h_1,h_2,e_1,e_2,f_1,f_2}. They satisfy the following relations.

1. [h_1,h_2]=0.

2. [e_1,f_1]=h_1 and [e_2,f_2]=h_2 while [e_1,f_2]=[e_2,f_1]=0.

3. [h_i,e_j]=A_(ij)e_j.

4. [h_i,f_j]=-A_(ij)f_j.

5. e_(12)=[e_1,e_2]!=0 and f_(12)=[f_1,f_2]!=0.

6. [e_i,e_(12)]=0 and [f_i,f_(12)]=0.

From these relations, it is not hard to see that g=sl_3 with the standard Lie algebra representation

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