Tetramodules Over a Bialgebra Form a 2-fold Monoidal Category

Abstract

Let B be an associative bialgebra over any field. A module over B in the sense of deformation theory is a tetramodule over B. All tetramodules form an abelian category. This category was studied by R. Taillefer (Algebr Represent Theory 7(5):471–490, 2004) and R. Taillefer (J Algebra 276(1):259–279, 2004). In particular, she proved that for any bialgebra B, the abelian category ${{\mathcal{T}}etra}(B)$ has enough injectives, and that Ext ∙ (B,B) in this category coincides with the Gerstenhaber-Schack cohomology of B. We prove that the category ${{\mathcal{T}}etra}(B)$ of tetramodules over any bialgebra B is a 2-fold-monoidal category, with B a unit object in it. Roughly, this means that the category ${{\mathcal{T}}etra}(B)$ admits two monoidal structures, with common unit B, which are compatible in some rather non-trivial way (the concept of an n-fold monoidal category is introduced in Baltenu et al. (Adv Math 176:277–349, 2003)). Within (yet unproven) 2-fold monoidal analogue of the Deligne conjecture, our result would imply that RHom ∙ (B,B) in the category of tetramodules is naturally a homotopy 3-algebra.