Hopf algebras, tetramodules, and n-fold monoidal categories

The abelian category of tetramodules over an associative bialgebra $A$ is related with the Gerstenhaber-Schack (GS) cohomology as $Ext_\Tetra(A,A)=H_\GS(A)$. We construct a 2-fold monoidal structure on the category of tetramodules of a bialgebra. Suppose $C$ is an abelian $n$-fold monoidal category with the unit object $A$. We prove, provided some condition (), that $Ext_C(A,A)$ is an $(n+1)$-algebra. In the case of bialgebras this condition () is satisfied when $A$ is a Hopf algebra. Finally, the GS cohomology of a Hopf algebra is a 3-algebra. As well, we consider this kind of questions of (bi)algebras over integers. Let $A$ be an associative algebra over $Z$ flat over $Z$. We prove that the operad acting on its Hochschild cohomology is the operad of stable homotopy groups of the little discs operad.

💡 Research Summary

The paper develops a new homological framework for associative bialgebras by introducing the category of tetramodules and showing how it encodes Gerstenhaber‑Schack (GS) cohomology. A tetramodule over a bialgebra A is an object equipped simultaneously with left and right A‑module structures and left and right A‑comodule structures that satisfy the natural compatibility conditions. The authors prove that the Ext‑groups in this abelian category satisfy
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