Semisimple Hopf Algebras via Geometric Invariant Theory

TL;DR

This paper employs geometric invariant theory to analyze semisimple Hopf algebras, establishing invariants that classify them, relate to automorphisms, and have implications for their orders and representation theory.

Contribution

It introduces a new invariant-theoretic framework for classifying semisimple Hopf algebras and connects these invariants to automorphisms, Frobenius-Schur indicators, and Kaplansky's conjecture.

Findings

Invariants determine Hopf algebra isomorphism classes.

Number of Hopf orders over a number ring is finite.

Invariants from Frobenius-Schur indicators relate to 3-manifold invariants.

Abstract

We study Hopf algebras via tools from geometric invariant theory. We show that all the invariants we get can be constructed using the integrals of the Hopf algebra and its dual together with the multiplication and the comultiplication, and that these invariants determine the isomorphism class of the Hopf algebra. We then define certain canonical subspaces $Inv^{i,j}$ of tensor powers of $H$ and $H^*$, and use the invariant theory to prove that these subspaces satisfy a certain non-degeneracy condition. Using this non-degeneracy condition together with results on symmetric monoidal categories, we prove that the spaces $Inv^{i,j}$ can also be described as $(H^{\otimes i}\otimes (H^*)^{\otimes j})^A$, where $A$ is the group of Hopf automorphisms of $H$. As a result we prove that the number of possible Hopf orders of any semisimple Hopf algebra over a given number ring is finite. we give some examples of these invariants arising from the theory of Frobenius-Schur Indicators, and from Reshetikhin-Turaev invariants of three manifolds. We give a complete description of the invariants for a group algebra, proving that they all encode the number of homomorphisms from some finitely presented group to the group. We also show that if all the invariants are algebraic integers, then the Hopf algebra satisfies Kaplansky's sixth conjecture: the dimensions of the irreducible representations of $H$ divide the dimension of $H$.