Semisimple Hopf algebras via geometric invariant theory

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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 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.
Original languageEnglish
Pages (from-to)61-90
Number of pages30
JournalAdvances in Mathematics
Volume311
Early online date28 Feb 2017
DOIs
Publication statusPublished - 30 Apr 2017

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Geometric Invariant Theory
Hopf Algebra
Semisimple
Invariant
Nondegeneracy
Subspace
Finitely Presented Groups
Algebraic integer
Three-manifolds
Monoidal Category
Invariant Theory
Isomorphism Classes
Group Algebra
Homomorphisms
Irreducible Representation
Divides
Automorphisms
Multiplication
Tensor
Ring

Keywords

  • Hopf algebras
  • Tensor categories
  • Symmetric monoidal categories
  • Geometric invariant theory
  • 3-manifolds invariants
  • Frobenius–Schur indicators

Cite this

Semisimple Hopf algebras via geometric invariant theory. / Meir, Ehud.

In: Advances in Mathematics, Vol. 311, 30.04.2017, p. 61-90.

Research output: Contribution to journalArticle

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