Quantizations of Kac-Moody Algebras

Numerous books and articles concerning quantizations of Kac-Moody algebras have been published. However, almost all publications have their own modifications in construction and different notations, so it is often unclear whether the results of one work may be applied to the construction of another. Nevertheless all of the constructions are character Hopf algebras. In view of the fact that the number and degrees of relations in all of the constructions related to a given KacMoody algebra g are identical, we introduce a class of character Hopf algebras defined by the same number of defining relations of the same degrees as the KacMoody algebra g is. This class contains all possible quantizations of g (including multiparameter quantizations), and these Hopf algebras are considered as quantum deformations of the universal enveloping algebra of g as well. The unification in the above class provides the potential to understand the differences, if any, between these constructions by comparing the basic invariants inside that class. We demonstrate that if the generalized Cartan matrix A of g is connected then the algebraic structure, up to a finite number of exceptional cases, is defined by just one "continuous" parameter q related to a symmetrization of A, and one "discrete" parameter m related to the modular symmetrizations of A. The Hopf algebra structure is defined by n(n 1)/2 additional "continuous" parameters.