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 Invi,j of tensor powers of H and H⁎, and use the invariant theory to prove that these subspaces satisfy a certain nondegeneracy condition. Using this nondegeneracy 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.
Original language  English 

Pages (fromto)  6190 
Number of pages  30 
Journal  Advances in Mathematics 
Volume  311 
Early online date  28 Feb 2017 
DOIs  
Publication status  Published  30 Apr 2017 
Bibliographical note
I 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”.Keywords
 Hopf algebras
 Tensor categories
 Symmetric monoidal categories
 Geometric invariant theory
 3manifolds invariants
 Frobenius–Schur indicators
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Ehud Meir Ben Efraim
 School of Natural & Computing Sciences, Mathematical Science  Senior Lecturer
Person: Academic