Pointed Hopf Algebras with Triangular Decomposition

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) 2014) 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 sl 2 .

On Hopf algebras with triangular decomposition

“…we will consider three suitable algebras which lead to a Hopf algebra with triangular decomposition. These kind of constructions were performed for instance in [17,23,28] in much more generality than we do it here, see §4.2.…”

“…4.2.2. In [23], Laugwitz analyzes how to endow T (V )⊗H⊗T (V * ) (and its quotients) with a structure of Hopf algebra. In particular, he extends the doublebosonization construction to a non-quasitriangular Hopf algebra H [23, §3.4].…”

“…In particular, he extends the doublebosonization construction to a non-quasitriangular Hopf algebra H [23, §3.4]. The data for his construction are two braided Hopf algebras B, C ∈ H H YD weakly dually paired [23,Definition 6].…”

On the Hopf algebra structure of the Lusztig quantum divided power algebras

“…We are not aware of other papers containing information on the matter of our interest. Other versions of triangular decompositions similar to [23] appear in [12,20].…”