Pointed Hopf Algebras with Triangular Decomposition: A Characterization of Multiparameter Quantum Groups

Laugwitz, Robert

Abstract

© 2016, The Author(s). 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 sl2.

Citation

Laugwitz, R. (2016). Pointed Hopf Algebras with Triangular Decomposition: A Characterization of Multiparameter Quantum Groups. Algebras and Representation Theory, 19(3), 547-578. https://doi.org/10.1007/s10468-015-9588-x

• Journal Article Type: Article
• Acceptance Date: Nov 13, 2015
• Online Publication Date: Mar 11, 2016
• Publication Date: 2016-06
• Deposit Date: Nov 28, 2019
• Publicly Available Date: Jan 20, 2020
• Journal: Algebras and Representation Theory
• Print ISSN: 1386-923X
• Electronic ISSN: 1572-9079
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 19
• Issue: 3
• Pages: 547-578
• DOI: https://doi.org/10.1007/s10468-015-9588-x
• Keywords: General Mathematics
• Public URL: https://nottingham-repository.worktribe.com/output/3440572
• Publisher URL: https://link.springer.com/article/10.1007%2Fs10468-015-9588-x
• Additional Information: Received: 23 August 2015; Accepted: 9 December 2015; First Online: 11 March 2016