Results 1  10
of
112
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 ..."
Abstract

Cited by 112 (6 self)
 Add to MetaCart
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. ..."
Abstract

Cited by 68 (5 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 65 (14 self)
 Add to MetaCart
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. ..."
Abstract

Cited by 63 (3 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 58 (3 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 51 (18 self)
 Add to MetaCart
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.
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. ..."
Abstract

Cited by 49 (7 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 36 (1 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 36 (4 self)
 Add to MetaCart
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.
Homotopy Gerstenhaber algebras and topological field theory, Operads
 Proceedings of Renaissance Conferences
, 1996
"... 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 mai ..."
Abstract

Cited by 27 (3 self)
 Add to MetaCart
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. Twodimensional topological quantum field theory (TQFT) at its most elementary level is the theory of Zgraded 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 Zgraded 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 Zgraded Poisson algebra for which the Poisson bracket has degree −1 (see Section 1). This Galgebra 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. Galgebras 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 Galgebras). 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 Galgebra, see Section 1; in fact, a double loop space is naturally