Exact Deformations of Quantum Groups; Applications to the Affine case

Abstract

This paper continues our investigation of a class of generalized quantum groups. The "standard" R-matrix was shown to be the unique solution of a very simple, linear recursion relation and the classical limit was obtained in the case of quantized Kac-Moody algebras of finite type. Here the standard R-matrix for generalized quantum groups is first examined in the case of quantized affine Kac-Moody algebras. The classical limit yields the standard affine r-matrices of Belavin and Drinfeld. Then, turning to the general case, we study the exact deformations of the standard R-matrix and the associated Hopf algebras. They are described as a generalized twist, $ R_ε= (F^t)^{-1}RF$, where $R$ is the standard R-matrix and $F$ (a power series in the deformation parameter $ε$) is the solution of a linear recursion relation of the same type as that which determines $R$. Specializing again, to the case of quantized, affine Kac-Moody algebras, and taking the classical limit of these esoteric quantum groups, one re-discovers the esoteric affine r-matrices of Belavin and Drinfeld, including the elliptic ones. The formulas obtained here are easier to use than the original ones, and the structure of the space of classical r-matrices (for simple Lie algebras) is more tranparent. In addition, the r-matrices obtained here are more general in that they are defined on the central extension of the loop groups.