Characteristic and Coxeter polynomials for affine Lie algebras

Abstract

We compute the characteristic polynomials of affine Cartan, adjacency matrices and Coxeter polynomials of the associated Coxeter system using Chebyshev polynomials. We give explicit factorization of these polynomials as products of cyclotomic polynomials. Finally, we present several different methods of obtaining the exponents and Coxeter number for affine Lie algebras. In particular we compute the exponents and Coxeter number for each conjugacy class in the case of $A_n^{(1)}$.