Quasi Hopf Deformations of Quantum Groups

The search for elliptic quantum groups leads to a modified quantum Yang-Baxter relation and to a special class of quasi-triangular quasi Hopf algebras. This paper calculates deformations of standard quantum groups (with or without spectral parameter) in the category of quasi-Hopf algebras. An earlier investigation of the deformations of quantum groups, in the category of Hopf algebras, showed that quantum groups are generically rigid: Hopf algebra deformations exist only under some restrictions on the parameters. In particular, affine Kac-Moody algebras are more rigid than their loop algebra quotients; and only the latter (in the case of sl(n)) can be deformed to elliptic Hopf algebras. The generalization to quasi-Hopf deformations lifts this restriction. The full elliptic quantum groups (with central extension) associated with $sl(n)$ are thus quasi-Hopf algebras. The universal R-matrices satisfy a modified Yang-Baxter relation and are calculated more or less explicitly. The modified classical Yang-Baxter relation is obtained, and the elliptic solutions are worked out explicitly. The same method is used to construct the Universal R-matrices associated with Felder's quantization of the Knizhnik-Zamolodchikov-Bernard equation, to throw some light on the quasi Hopf structure of conformal field theory and (perhaps) the Calogero-Moser models.