Abstract
In this paper, we present an approach to the definition of multiparameter quantum groups by studying Hopf algebras with triangular decomposition. Classifying all of these Hopf algebras which are of what we call weakly separable type over a group, we obtain a class of pointed Hopf algebras which can be viewed as natural generalizations of multiparameter deformations of universal enveloping algebras of Lie algebras. These Hopf algebras are instances of a new version of braided Drinfeld doubles, which we call asymmetric braided Drinfeld doubles. This is a generalization of an earlier result by Benkart and Witherspoon (Algebr. Represent. Theory 7(3) ? BC) who showed that two-parameter quantum groups are Drinfeld doubles. It is possible to recover a Lie algebra from these doubles in the case where the group is free abelian and the parameters are generic. The Lie algebras arising are generated by Lie subalgebras isomorphic to $\mathfrak {sl}_{2}$ .
Highlights
1.1 What are Quantum Groups?An important problem in the theory of quantum groups is to give some definition of a class of these objects that captures known series of quantum groups, such as the quantum enveloping algebras Uq (g) of [18], and their finite-dimensional analogues, as examples
We aim to provide an axiomatic approach to the definition of quantum groups by combining the pointed Hopf algebra and the triangular decomposition approach
It was understood early that some pointed Hopf algebras can be obtained as bosonizations
Summary
An important problem in the theory of quantum groups is to give some definition of a class of these objects that captures known series of quantum groups, such as the quantum enveloping algebras Uq (g) of [18], and their finite-dimensional analogues, as examples. A possible hint to the structure of quantum groups is that the quantum enveloping algebras Uq (g) (as well as the small quantum groups uq (g) and multiparameter versions) are pointed Hopf algebras. A third aspect — observed already in the original paper [18] — is that quantum groups are (quotients of) quantum or Drinfeld doubles It was shown in [24] that Uq (g) is a braided Drinfeld double (which is referred to as a double bosonization there). Under certain assumptions on the group and the parameters, we can recover Lie algebras from these Hopf algebras, after introducing a suitable integral form
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