Solutions of Discretized Affine Toda Field Equations for A_{n}^{(1)}, B_{n}^{(1)}, C_{n}^{(1)}, D_{n}^{(1)}, A_{n}^{(2)} and D_{n+1}^{(2)}

Abstract

It is known that a family of transfer matrix functional equations, the T-system, can be compactly written in terms of the Cartan matrix of a simple Lie algebra. We formally replace this Cartan matrix of a simple Lie algebra with that of an affine Lie algebra, and then we obtain a system of functional equations different from the T-system. It may be viewed as an X_{n}^{(a)} type affine Toda field equation on discrete space time. We present, for A_{n}^{(1)}, B_{n}^{(1)}, C_{n}^{(1)}, D_{n}^{(1)}, A_{n}^{(2)} and D_{n+1}^{(2)}, its solutions in terms of determinants or Pfaffians.