We show that the Connes-Moscovici cyclic cohomology of a Hopf algebra equipped with a character has a Lie bracket of degree −2. More generally, we show that a "cyclic operad with multiplication" is a cocyclic module whose cohomology is a Batalin-Vilkovisky algebra and whose cyclic cohomology is a graded Lie algebra of degree −2. This explain why the Hochschild cohomology algebra of a symmetric algebra is a Batalin-Vilkovisky algebra.1991 Mathematics Subject Classification. 16W30, 19D55, 16E40, 18D50.

Abstract. Let G be a finite group or a compact connected Lie group and let BG be its classifying space. Let LBG := map(S 1 , BG) be the free loop space of BG i.e. the space of continuous maps from the circle S 1 to BG. The purpose of this paper is to study the singular homology H * (LBG) of this loop space. We prove that when taken with coefficients in a field the homology of LBG is a homological conformal field theory. As a byproduct of our main theorem, we get a Batalin-Vilkovisky algebra structure on the cohomology H * (LBG). We also prove an algebraic version of this result by showing that the Hochschild cohomology HH

Let C be a differential graded chain coalgebra, C the reduced cobar construction on C and C ∨ the dual algebra. We prove that for a large class of coalgebras C there is a natural isomorphism of Gerstenhaber algebras between the Hochschild cohomologies HH * (C ∨ , C ∨ ) and HH * ( C; C). This result yields to a Hodge decomposition of the loop space homology of a closed oriented manifold, when the field of coefficients is of characteristic zero.

Abstract. Let M be a compact oriented d-dimensional smooth manifold. Chas and Sullivan have defined a structure of BatalinVilkovisky algebra on H * (LM ). Extending work of Cohen, Jones and Yan, we compute this Batalin-Vilkovisky algebra structure when M is a sphere S d , d ≥ 1. In particular, we show that H * (LS 2 ; F 2 ) and the Hochschild cohomology HH * (H * (S 2 ); H * (S 2 )) are surprisingly not isomorphic as Batalin-Vilkovisky algebras, although as expected the underlying Gerstenhaber algebras are isomorphic.