# Generalization and Exact Deformations of Quantum Groups

### Christian Frønsdal

University of California, Los Angeles, United States

## Abstract

A large family of "standard" coboundary Hopf algebras is investigated. The existence of a universal R-matrix is demonstrated for the case when the parameters are in general position. Algebraic surfaces in parameter space are characterized by the appearance of certain ideals; in this case the universal R-matrix exists on the associated algebraic quotient. In special cases the quotient is a "standard" quantum group; all familiar quantum groups including twisted ones are obtained in this way. In other special cases one finds new types of coboundary bi-algebras. The "standard" universal R-matrix is shown to be the unique solution of a very simple, linear recursion relation. The classical limit is obtained in the case of quantized Kac-Moody algebras of finite and affine type. Returning to the general case, we study deformations of the standard R-matrix and the associated Hopf algebras. A preliminary investigation of the first order deformations uncovers a class of deformations that incompasses the quantization of all Kac-Moody algebras of finite and affine type. The corresponding exact deformations are described as generalized twists, *R_∊ = ( Ft)–1_RF*, where

*R*is the standard R-matrix and the cocycle

*F*(a power series in the deformation parameter ∊) is the solution of a linear recursion relation of the same type as that which determines

*R*. Included here is the universal R-matrix for the elliptic quantum groups associated with

*sl*(

*n*), a big surprise! Specializing again, to the case of quantized Kac-Moody algebras, and taking the classical limit of these esoteric quantum groups, one re-discovers all the trigonometric and elliptic r-matrices of Belavin and Drinfeld. The formulas obtained here are easier to use than the original ones, and the structure of the space of classical r-matrices is more transparent. The r-matrices obtained here are more general in that they are defined on the full Kac-Moody algebras, the central extensions of the loop groups.

## Cite this article

Christian Frønsdal, Generalization and Exact Deformations of Quantum Groups. Publ. Res. Inst. Math. Sci. 33 (1997), no. 1, pp. 91–149

DOI 10.2977/PRIMS/1195145535