Serre Theorem for Involutory Hopf Algebras
نویسنده
چکیده
Let H be an involutory Hopf algebra over a field of characteristic zero, M and N two finite dimensional left H-modules such that M ⊗ N is a semisimple H-module. Then M and N are semisimple H-modules. This is a generalization of a theorem proved by J.-P. Serre for group algebras. A version of the theorem above for monoidal categories is also given.
منابع مشابه
The Equality of 3-manifold Invariants
The invariants of 3-manifolds defined by Kuperberg for involutory Hopf algebras and those defined by the authors for spherical Hopf algebras are the same for Hopf algebras on which they are both defined. Introduction The purpose of this paper is to compare two previously defined invariants of 3-manifolds. Let A be a finite-dimensional Hopf algebra over a field F with antipode S. Then if S = 1 t...
متن کاملar X iv : h ep - t h / 93 10 16 4 v 2 2 2 Ju l 1 99 8 SPHERICAL CATEGORIES
This paper is a study of monoidal categories with duals where the tensor product need not be commutative. The motivating examples are categories of representations of Hopf algebras. We introduce the new notion of a spherical category. In the first section we prove a coherence theorem for a monoidal category with duals following [MacLane 1963]. In the second section we give the definition of a s...
متن کاملSerre-Swan theorem for non-commutative C∗-algebras. Revised edition
We generalize the Serre-Swan theorem to non-commutative C∗algebras. For a Hilbert C∗-module X over a C∗-algebra A, we introduce a hermitian vector bundle EX associated to X . We show that there is a linear subspace ΓX of the space of all holomorphic sections of EX and a flat connection D on EX with the following properties: (i) ΓX is a Hilbert A-module with the action of A defined by D, (ii) th...
متن کاملCosovereign Hopf algebras
A sovereign monoidal category is an autonomous monoidal category endowed with the choice of an autonomous structure and an isomorphism of monoidal functors between the associated left and right duality functors. In this paper we define and study the algebraic counterpart of sovereign monoidal categories: cosovereign Hopf algebras. In this framework we find a categorical characterization of invo...
متن کاملInvolutory Hopf Group-coalgebras and Flat Bundles over 3-manifolds
Given a group π, we use involutary Hopf π-coalgebras to define a scalar invariant of flat π-bundles over 3-manifolds. When π = 1, this invariant equals to the one of 3-manifolds constructed by Kuperberg from involutary Hopf algebras. We give examples which show that this invariant is not trivial.
متن کامل