Lecture notes for Introduction to Representation Theory

5.4 Roots
From now on, let be a fixed graph of type A n , D n , E 6 , E 7 , E 8 . We denote the adjacency matrix
of by R .
Definition 5.11 (Cartan Matrix). We define the Cartan matrix as
A = 2Id − R .
On the lattice Z n (or the space R n ) we then define an inner product
corresponding to the graph .
B(x, y) = x T A y
Lemma 5.12.
1. B is positive definite.
2. B(x, x) takes only even values for x Z n .
Proof.
1. This follows by definition, since is a Dynkin diagram.
2. By the definition of the Cartan matrix we get
2 2
B(x, x) = x T A x = x i a ij x j = 2 x i + x i a ij x j = 2 x i + 2 ·
a ij x i x j
which is even.
i,j i i,j, i=j ∞
i i 0, k j < 0. Without loss of generality, we can also assume
that k s = 0 for all s between i and j. We can identify the indices i, j with vertices of the graph .
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