Tetramodules over a bialgebra form a 2-fold monoidal category

TL;DR

This paper constructs a 2-fold monoidal structure on the category of tetramodules over a bialgebra, reviews related cohomology results, and computes Gerstenhaber-Schack cohomology for a specific class of bialgebras.

Contribution

It introduces a 2-fold monoidal structure on tetramodules and provides explicit cohomology computations for free commutative cocommutative bialgebras.

Findings

Construction of 2-fold monoidal structure on tetramodules

Overview of Gerstenhaber-Schack cohomology results

Explicit cohomology computation for S(V)

Abstract

This preprint contains a part of the results of our earlier preprint arXiv:0907.3335v2 presented in a form suitable for journal publication. It covers a construction of a 2-fold monoidal structure on the category of tetramodules, with all necessary definitions, and an overview of the results of R.Taillefer [Tai1,2] on tetramodules and the Gerstenhaber-Schack cohomology [GS] (formerly served as Appendix in arXiv:0907.3335v2), as well as a computation of the Gerstenhaber-Schack cohomology for the free commutative cocommutative bialgebra S(V), for a V is a vector space.