On algebraic structures of the Hochschild complex
Résumé
We review various algebraic structures on the Hochschild homology and cohomology of a differential graded algebra $A$ under a weak Poincaré duality hypothesis. This includes a BV-algebra structure on $HH^*(A,A^\vee)$ or $HH^*(A,A)$, which in the latter case is an extension of the natural Gerstenhaber structure on $HH^*(A,A)$. In sections 6 and 7 we construct similar structures for open Frobenius DG-algebras. In particular we prove that the Hochschild homology and cohomology of an open Frobenius algebra is a BV-algebra. In other words we prove that Hochschild chains complex is homotopical BV and coBV algebra. In Section 7 we present an action of the Sullivan diagrams on the Hochschild (co)chain complex of an open Frobenius DG-algebra. This recovers Tradler-Zeinalian \cite{TZ} result for closed Froebenius algebras using the isomorphism $C^*(A ,A) \simeq C^*(A,A^\vee)$. Our description of the action can be easily and without much of modification extended to the homotopy Frobenius algebras.
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