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111
Deformations of algebras over operads and the Deligne conjecture, Conférence Moshé Flato
, 1999
"... The deformation theory of associative algebras is a guide for developing the deformation theory of many algebraic structures. Conversely, all the ..."
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Cited by 112 (6 self)
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The deformation theory of associative algebras is a guide for developing the deformation theory of many algebraic structures. Conversely, all the
Modular Operads
 COMPOSITIO MATH
, 1994
"... 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. ..."
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Cited by 70 (5 self)
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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.
PreLie algebras and the rooted trees operad
 Internat. Math. Res. Notices
"... Abstract. A preLie 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 o ..."
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Cited by 64 (13 self)
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Abstract. A preLie 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.
On differential graded categories
 INTERNATIONAL CONGRESS OF MATHEMATICIANS. VOL. II
, 2006
"... 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, DuggerShipley,..., Toën and ToënVaquié. ..."
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Cited by 63 (3 self)
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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, DuggerShipley,..., Toën and ToënVaquié.
A.: Homotopy Galgebras and moduli space operad
 Preprint MPI / 9471, MaxPlanckInstitut in Bonn
, 1994
"... hepth/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 cocha ..."
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Cited by 57 (3 self)
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hepth/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 Galgebra structure on Ω • X. This latter statement is based on a typical construction of supersymmetric sigmamodel, the construction of GromovWitten 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 GromovWitten 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
Axiomatic Homotopy Theory for Operads
 Comment. Math. Helv
, 2002
"... 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. ..."
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Cited by 52 (7 self)
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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.
Combinatorial operad actions on cochains
, 2001
"... A classical Einfinity 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 FKconstruction for infinite loop spaces. The purpose of this article i ..."
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Cited by 51 (18 self)
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A classical Einfinity 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 FKconstruction 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 BarrattEccles operad. We also prove that differential graded algebras over the BarrattEccles 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 BarrattEccles operad which is equivalent to the classical little squares operad.
Topological Open pBranes
, 2000
"... By exploiting the BV quantization of topological bosonic open membrane, we argue that flat 3form Cfield leads to deformations of the algebras of multivectors on the Dirichletbrane worldvolume as 2algebras. This would shed some new light on geometry of Mtheory 5brane and associated decoupled ..."
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Cited by 37 (1 self)
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By exploiting the BV quantization of topological bosonic open membrane, we argue that flat 3form Cfield leads to deformations of the algebras of multivectors on the Dirichletbrane worldvolume as 2algebras. This would shed some new light on geometry of Mtheory 5brane and associated decoupled theories. We show that, in general, topological open pbrane has a structure of (p + 1)algebra in the bulk, while a structure of palgebra in the boundary. The bulk/boundary correspondences are exactly as of the generalized Deligne conjecture (a theorem of Kontsevich) in the algebraic world of palgebras. It also imply that the algebras of quantum observables of (p − 1)brane are “close to ” the algebras of its classical observables as palgebras. We interpret above as deformation quantization of (p − 1)brane, generalizing the p = 1 case. We argue that there is such quantization based on the direct relation between BV master equation and Ward identity of the bulk topological theory. The path integral of the theory will lead to the explicit formula. We also discuss some applications to
On the structure of cofree Hopf algebras
 J. reine angew. Math
"... Abstract. We prove an analogue of the PoincaréBirkhoffWitt theorem and of the CartierMilnorMoore theorem for noncocommutative 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∞algebr ..."
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Cited by 33 (4 self)
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Abstract. We prove an analogue of the PoincaréBirkhoffWitt theorem and of the CartierMilnorMoore theorem for noncocommutative 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 2associative 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 2asalgebra in terms of planar trees to unravel the operad associated to nondifferential B∞algebras.
Cyclic operads and cyclic homology
 in "Geometry, Topology and Physics,"International
, 1995
"... 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 noncommutative setting. Recently, there has been increased interest in more general algebraic structures than associative algebras, characte ..."
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Cited by 27 (2 self)
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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 noncommutative 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 nary 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 Snmodules, 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