On algebraic structures of the Hochschild complex

Hossein Abbaspour (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray

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.

Mots clés: Hochschild complex; string topology; Calabi-Yau algebras; Frobenius algebras; Poincaré duality; Sullivan diagrams; module space of curves; Batalin-Vilkovisky algebras; Gerstenhaber algebras

Domaines: Topologie algébrique [math.AT]; Algèbres quantiques [math.QA]; Géométrie algébrique [math.AG]

Hossein Abbaspour. On algebraic structures of the Hochschild complex. 2010. ⟨hal-00794525v1⟩