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Cartan Matrix


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

h_1=[ 1  0  0; 0  -1  0; 0  0  0]
(4)
h_2=[ 0  0  0; 0  1  0; 0  0  -1]
(5)
e_1=[ 0  1  0; 0  0  0; 0  0  0]
(6)
e_2=[ 0  0  0; 0  0  1; 0  0  0]
(7)
e_(12)=[ 0  0  1; 0  0  0; 0  0  0]
(8)
f_1=[ 0  0  0; 1  0  0; 0  0  0]
(9)
f_2=[ 0  0  0; 0  0  0; 0  1  0]
(10)
f_(12)=[ 0  0  0; 0  0  0; -1  0  0].
(11)

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


See also

Dynkin Diagram, Lie Algebra, Lie Algebra Root, Root System, Semisimple Lie Algebra, Weyl Group

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Cartan Matrix." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/CartanMatrix.html

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