Affine Quantum Groups

Abstract

(0.1) R12(u)R13(u + v)R23(v) = R23(v)R13(u + v)R12(u). One can distinguish three classes of affine quantum groups, each leading to a different dependence of the R-matrices on the spectral parameter u: Yangians lead to rational R-matrices, quantum affine algebras lead to trigonometric R-matrices and elliptic quantum groups lead to elliptic R-matrices. We will mostly concentrate on the quantum affine algebras but many results hold similarly for the other c After giving mathematical details about quantum affine algebras and Yangians in the first two section, we describe how these algebras arise in different areas of mathematical physics in the three following sections. We end with a description of boundary quantum groups which extend the formalism to the boundary Yang-Baxter (reflection) equation.