Restricted quantum groups as graded Hopf algebras

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.

51 pages