# Hopf algebras, tetramodules, and n-fold monoidal categories

### Boris Shoikhet (U Luxembourg)

Wednesday 27th October, 2010 16:00-17:00 204

#### Abstract

Let $M$ be an abelian monoidal category, with some "homotopy exactness" property of the monoidal structure. Then for the unit object $e$ in $M$ the graded space $Ext^(e,e)$ is a commutative graded algebra with a Lie bracket of degree -1 on it, the two structures are compatible by the Leibniz rule. This structure is called a Gerstenhaber algebra, or a 2-algebra. An example of such situation is $Ext^(A,A)$ in the category of $A$-bimodules over an associative algebra $A$, which gives the Hochschild cohomology of $A$. We will discuss the general theorem above, as well as its generalization for so called $n$-fold monoidal categories. We prove that the corresponding space of $Ext^(e,e)$ in the $n$-monoidal case is an $(n+1)$-algebra. We give an example for $n=2$, the category of tetramodules over an associative bialgebra $A$. Then our theorem in this case states that the Gerstenhaber-Schack cohomology of a Hopf algebra $A$ is a 3-algebra, confirming a conjecture of M.Kontsevich.

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