Generalization and Exact Deformations of Quantum Groups

Details Generalization and Exact Deformations of Quantum Groups Generalization and Exact Deformations of Quantum Groups

1996-06-25

arxiv.org/abs/q-alg/9606020v1

Pub. Res. Inst. Math. Sci. 33 (1997) 91-149

Mathematics

Quantum Algebra

62 pages, plain Tex. This paper combines q-alg/9510002 and q-alg/9602034; Submitted for publication

Scientific paper

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_\epsilon = (F^t)^{-1}RF$, where $R$ is the standard R-matrix and the cocycle $F$ (a power series in the deformation parameter $\epsilon$) 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.

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