A Categorical Approach to Subgroups of Quantum Groups and Their Crystal Bases

TL;DR

This paper explores the categorical structure of subgroups within quantum groups and their crystal bases, establishing a framework that links quantum subgroup theory with crystal basis theory.

Contribution

It introduces a categorical approach to subgroups of quantum groups, defining subgroups as right coideal subalgebras and quotient coalgebras, connecting quantum groups with crystal bases.

Findings

The category of crystals for $U_q(rak{g})$ is monoidal.

Subgroups are characterized as right coideal subalgebras and quotient coalgebras.

The approach links quantum subgroup structures to crystal basis theory.

Abstract

Suppose that we have a semisimple, connected, simply connected algebraic group $G$ with corresponding Lie algebra $\mathfrak{g}$. There is a Hopf pairing between the universal enveloping algebra $U(\mathfrak{g})$ and the coordinate ring $O(G)$. By introducing a parameter $q$, we can consider quantum deformations $U_q(\mathfrak{g})$ and $O_q(G)$ respectively, between which there again exists a Hopf pairing. We show that the category of crystals associated with $U_q(\mathfrak{g})$ is a monoidal category. We define subgroups of $U_q(\mathfrak{g})$ to be right coideal subalgebras, and subgroups of $O_q(G)$ to be quotient left $O_q(G)$-module coalgebras. Furthermore, we discuss a categorical approach to subgroups of quantum groups which we hope will provide us with a link to crystal basis theory.