The deformation theory of associative algebras is a guide for developing the deformation theory of many algebraic structures. Conversely, all the

We develop a "higher genus" analogue of operads, which we call modular operads, in which graphs replace trees in the definition. We study a functor F on the category of modular operads, the Feynman transform, which generalizes Kontsevich's graph complexes and also the bar construction for operads. We calculate the Euler characteristic of the Feynman transform, using the theory of symmetric functions: our formula is modelled on Wick's theorem. We give applications to the theory of moduli spaces of pointed algebraic curves.

A classical E-infinity operad is formed by the bar construction of the symmetric groups. Such an operad has been introduced by M. Barratt and P. Eccles in the context of simplicial sets in order to have an analogue of the Milnor FK-construction for infinite loop spaces. The purpose of this article is to prove that the associative algebra structure on the normalized cochain complex of a simplicial set extends to the structure of an algebra over the Barratt-Eccles operad. We also prove that differential graded algebras over the Barratt-Eccles operad form a closed model category. Similar results hold for the normalized Hochschild cochain complex of an associative algebra. More precisely, the Hochschild cochain complex is acted on by a suboperad of the Barratt-Eccles operad which is equivalent to the classical little squares operad.

hep-th/9409063 Abstract. This paper emphasizes the ubiquitous role of moduli spaces of algebraic curves in associative algebra and algebraic topology. The main results are: (1) the space of an operad with multiplication is a homotopy G- (i.e., homotopy graded Poisson) algebra; (2) the singular cochain complex is naturally an operad; (3) the operad of decorated moduli spaces acts naturally on the de Rham complex Ω • X of a Kähler manifold X, thereby yielding the most general type of homotopy G-algebra structure on Ω • X. This latter statement is based on a typical construction of supersymmetric sigma-model, the construction of Gromov-Witten invariants in Kontsevich’s version. Recently, shortly after a conjecture of Deligne [4], new algebraic structures on the Hochschild cochain space V = C • (A, A) of an associative algebra have been discovered [8, 6]. It has also been pointed out in [8] that a similar structure takes place for the singular cochain complex V = C • X of a topological space, due to Baues [2]. In this paper, we find a very general pattern which works for these two examples: in both cases, V has a natural structure of an operad. Together with a multiplication, it yields all the complicated algebraic buildup on V, see Sections 1 and 2. The rest of the paper is dedicated to the geometry of the conjecture, which, in fact, assumed something more than mere algebraic structure: Conjecture (Deligne). The Hochschild cochain complex has a natural structure of an algebra over a chain operad of the little squares operad. In Section 4.2, we use the construction of Gromov-Witten invariants by Kontsevich to propose a way of proving the result analogous to the conjecture in the case of singular cochain complex. Acknowledgment. We thank J. Stasheff for helpful comments on an early version of the manuscript. The second author would like to thank

We give sufficient conditions for the existence of a model structure on operads in an arbitrary symmetric monoidal model category. General invariance properties for homotopy algebras over operads are deduced.

Abstract. A pre-Lie algebra is a vector space L endowed with a bilinear product · : L × L → L satisfying the relation (x · y) · z − x · (y · z) = (x · z) · y − x · (z · y), ∀x, y, z ∈ L. We give an explicit combinatorial description in terms of rooted trees of the operad associated to this type of algebras and prove that it is a Koszul operad.

Abstract. Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, Dugger-Shipley,..., Toën and Toën-Vaquié. 1.

Abstract. We prove that the BRST complex of a topological conformal field theory is a homotopy Gerstenhaber algebra, as conjectured by Lian and Zuckerman in 1992. We also suggest a refinement of the original conjecture for topological vertex operator algebras. We illustrate the usefulness of our main tools, operads and “string vertices ” by obtaining new results on Vassiliev invariants of knots and double loop spaces. Two-dimensional topological quantum field theory (TQFT) at its most elementary level is the theory of Z-graded commutative associative algebras (with some additional structure) [34]. Thus, it came as something of a surprise when several groups of mathematicians realized that the physical state space of a 2D TQFT has the structure of a Z-graded Lie algebra, relative to a new grading equal to the old grading minus one. Moreover, the commutative and Lie products fit together nicely to give the structure of a Gerstenhaber algebra (Galgebra), a Z-graded Poisson algebra for which the Poisson bracket has degree −1 (see Section 1). This G-algebra structure is best understood in the framework of 2D topological conformal field theories (TCFTs) (see Section 5.2) wherein operads of moduli spaces of Riemann surfaces play a fundamental role. G-algebras arose explicitly in M. Gerstenhaber’s work on the Hochschild cohomology theory for associative algebras (see Section 1 for this and several other contexts for the theory of G-algebras). Operads arose in the work of J. Stasheff, Gerstenhaber and later work of P. May on the recognition problem for iterated loop spaces. Eventually, F. Cohen discovered that the homology of a double loop space is naturally a G-algebra, see Section 1; in fact, a double loop space is naturally

Abstract. We prove an analogue of the Poincaré-Birkhoff-Witt theorem and of the Cartier-Milnor-Moore theorem for non-cocommutative Hopf algebras. The primitive part of a cofree Hopf algebra is a nondifferential B∞-algebra. We construct a universal enveloping functor U2 from nondifferential B∞-algebras to 2-associative algebras, i.e. algebras equipped with two associative operations. We show that any cofree Hopf algebra H is of the form U2(Prim H). We take advantage of the description of the free 2as-algebra in terms of planar trees to unravel the operad associated to nondifferential B∞-algebras.

The cyclic homology of associative algebras was introduced by Connes [4] and Tsygan [22] in order to extend the classical theory of the Chern character to the non-commutative setting. Recently, there has been increased interest in more general algebraic structures than associative algebras, characterized by the presence of several algebraic operations. Such structures appear, for example, in homotopy theory [18], [3] and topological field theory [9]. In this paper, we extend the formalism of cyclic homology to this more general framework. This extension is only possible under certain conditions which are best explained using the concept of an operad. In this approach to universal algebra, an algebraic structure is described by giving, for each n ≥ 0, the space P(n) of all n-ary expressions which can be formed from the operations in the given algebraic structure, modulo the universally valid identities. Permuting the arguments of the expressions gives an action of the symmetric group Sn on P(n). The sequence P = {P(n)} of these Sn-modules, together with the natural composition structure on them, is the operad describing our class of algebras. In order to define cyclic homology for algebras over an operad P, it is necessary that P is what we call a cyclic operad: this means that the action of Sn on P(n) extends to an action of Sn+1 in a way compatible with compositions (see Section 2). Cyclic