Restricted quantum groups as graded Hopf algebras
We introduce the notion of $π^2$-graded Hopf algebra, where the grading is by the double groupoid of commutative diagrams of a finite groupoid $π$. The finite dimensional representations of a $π^2$-graded Hopf algebra form a rigid monoidal category with a fibre functor to the category of $π$-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.
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