  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 $A_n^{(1)}$.
