Restricted quantum groups as graded Hopf algebras

TL;DR

This paper introduces $pi^2$-graded Hopf algebras, linking their representation categories to models in statistical mechanics and Lie algebra theory.

Contribution

It defines $pi^2$-graded Hopf algebras and connects their representation theory to well-known models in statistical mechanics and Lie algebra.

Findings

Finite dimensional representations form a rigid monoidal category.

Main example is the restricted quantum groups related to statistical models.

Connects graded Hopf algebras to classical Lie algebra models.

Abstract

We introduce the notion of $\pi^2$-graded Hopf algebra, where the grading is by the double groupoid of commutative diagrams of a finite groupoid $\pi$. The finite dimensional representations of a $\pi^2$-graded Hopf algebra form a rigid monoidal category with a fibre functor to the category of $\pi$-graded vector spaces. The main example is given by the restricted quantum groups underlying the Andrews-Baxter-Forrester restricted solid-on-solid models of statistical mechanics and, more generally, the Jimbo-Miwa-Okado models associated to classical Lie algebras.