Lecture notes for Introduction to Representation Theory
Lecture notes for Introduction to Representation Theory
Lecture notes for Introduction to Representation Theory
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5.4 Roots<br />
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<br />
of by R .<br />
Definition 5.11 (Cartan Matrix). We define the Cartan matrix as<br />
A = 2Id − R .<br />
On the lattice Z n (or the space R n ) we then define an inner product<br />
corresponding <strong>to</strong> the graph .<br />
B(x, y) = x T A y<br />
Lemma 5.12.<br />
1. B is positive definite.<br />
2. B(x, x) takes only even values <strong>for</strong> x Z n .<br />
Proof.<br />
1. This follows by definition, since is a Dynkin diagram.<br />
2. By the definition of the Cartan matrix we get<br />
2 2<br />
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 ·<br />
a ij x i x j<br />
which is even.<br />
i,j i i,j, i=j ∞<br />
i i 0, k j < 0. Without loss of generality, we can also assume<br />
that k s = 0 <strong>for</strong> all s between i and j. We can identify the indices i, j with vertices of the graph .<br />
• •<br />
i<br />
ρ<br />
i <br />
• • • j<br />
• •<br />
•<br />
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