Multi-parameter Quantum Groups and Quantum Shuffles, (I)

TL;DR

This paper explores multi-parameter quantum groups linked to symmetrizable Cartan matrices, providing explicit descriptions as Hopf 2-cocycle deformations and quantum shuffle realizations, enhancing understanding of their structure and representations.

Contribution

It introduces explicit descriptions of multi-parameter quantum groups as Hopf 2-cocycle deformations and quantum shuffle realizations, facilitating further study and applications.

Findings

Explicit Hopf 2-cocycle deformation description

Quantum shuffle realization of the positive part

Framework for studying representations in category $\

Abstract

In this article, we study the multi-parameter quantum groups defined by generators and relations associated with symmetrizable generalized Cartan matrices, together with their representations in the category $\mathcal O$. This presentation will be convenient for our later discussions. We present two explicit descriptions here: as a Hopf 2-cocycle deformation, and as the multi-parameter quantum shuffle realization of the positive part.