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 Invi,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 Invi,j can also be described as (H⊗i⊗(H⁎)⊗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.
Bibliographical noteI first encountered Geometric Invariant Theory during a program on moduli spaces at the Isaac Newton Institute in Cambridge at the first half of 2011. I would like to thank the Newton Institute and the organizers of the program. I would also like to thank the referee for carefully reading the manuscript and for some valuable comments. During the writing of this paper I was supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92) and by the Research Training Group1670 “Mathematics inspired by String theory and Quantum Field Theory”.
- Hopf algebras
- Tensor categories
- Symmetric monoidal categories
- Geometric invariant theory
- 3-manifolds invariants
- Frobenius–Schur indicators