Generalization and Exact Deformations of Quantum Groups

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, $R_{ϵ}=(F^t{)}^{–1}RF$, where $R$ is the standard R-matrix and the cocycle $F$ (a power series in the deformation parameter $ϵ$) is the solution of a linear recursion relation of the same type as that which determines $R$. Included here is the universal R-matrix for the elliptic quantum groups associated with $\text{sl}(n)$, 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.