Results 1  10
of
24
Homology of generalized partition posets
 Journal of Pure and Applied Algebra, Volume 208, Issue
, 2007
"... Abstract. We define a family of posets of partitions associated to an operad. We prove that the operad is Koszul if and only if the posets are CohenMacaulay. On the one hand, this characterization allows us to compute completely the homology of the posets. The homology groups are isomorphic to the ..."
Abstract

Cited by 20 (4 self)
 Add to MetaCart
Abstract. We define a family of posets of partitions associated to an operad. We prove that the operad is Koszul if and only if the posets are CohenMacaulay. On the one hand, this characterization allows us to compute completely the homology of the posets. The homology groups are isomorphic to the Koszul dual cooperad. On the other hand, we get new methods for proving that an operad is Koszul.
Homotopy Gerstenhaber algebras
 OF SCIENCES OF THE CZECH REPUBLIC MATHEMATICAL INSTITUTE
, 2000
"... Dedicated to the memory of Moshé Flato Abstract. The purpose of this paper is to complete GetzlerJones ’ proof of Deligne’s Conjecture, thereby establishing an explicit relationship between the geometry of configurations of points in the plane and the Hochschild complex of an associative algebra. M ..."
Abstract

Cited by 19 (0 self)
 Add to MetaCart
Dedicated to the memory of Moshé Flato Abstract. The purpose of this paper is to complete GetzlerJones ’ proof of Deligne’s Conjecture, thereby establishing an explicit relationship between the geometry of configurations of points in the plane and the Hochschild complex of an associative algebra. More concretely, it is shown that the B∞operad, which is generated by multilinear operations known to act on the Hochschild complex, is a quotient of a certain operad associated to the compactified configuration spaces. Different notions of homotopy Gerstenhaber algebras are discussed: One of them is a B∞algebra, another, called a homotopy Galgebra, is a particular case of a B∞algebra, the others, a G∞algebra, an E 1algebra, and a weak G∞algebra, arise from the geometry of configuration spaces. Corrections to the paper of Kimura, Zuckerman, and the author related to the use of a nonextant notion of a homotopy Gerstenhaber algebra are made. In an unpublished paper of E. Getzler and J. D. S. Jones [GJ94], the notion of a homotopy nalgebra was introduced. Unfortunately the construction that justified
Prop profile of deformation quantization and graph complexes with loops and wheels
"... Motivated by the problem of deformation quantization we introduce and study directed graph complexes with oriented loops and wheels. We develop a new technique for computing cohomology of such graph complexes in terms of other much simpler purely operadic graph complexes. As an application we comput ..."
Abstract

Cited by 15 (4 self)
 Add to MetaCart
Motivated by the problem of deformation quantization we introduce and study directed graph complexes with oriented loops and wheels. We develop a new technique for computing cohomology of such graph complexes in terms of other much simpler purely operadic graph complexes. As an application we compute cohomology of wheeled extensions of several classical examples (including the one associated with the minimal resolution of the PROP of Lie bialgebras) and also give a new purely PROPic proof of Kontsevich’s theorem on existence of star products on formal germs of Poisson manifolds. The first instances of graph complexes have been introduced in the theory of operads and PROPs which have found recently lots of applications in algebra, topology and geometry. Another set of examples has been introduced by Kontsevich [Ko1, Ko2] as a way to expose highly nontrivial interrelations between certain infinite dimensional Lie algebras and topological
Profile of Poisson Geometry
 Comm. Math.Phys
"... “The genetic code appears to be universal;... ” ..."
Character formulas for the operad of two compatible brackets and for the bihamiltonian operad
"... Abstract. We compute dimensions of the components for the operad of two compatible brackets and for the bihamiltonian operad. We also obtain character formulas for the representations of the symmetric groups and the SL2 group in these spaces. 1. ..."
Abstract

Cited by 10 (4 self)
 Add to MetaCart
Abstract. We compute dimensions of the components for the operad of two compatible brackets and for the bihamiltonian operad. We also obtain character formulas for the representations of the symmetric groups and the SL2 group in these spaces. 1.
Framed discs operads and the equivariant recognition principle
"... The framed ndiscs operad fDn is studied as semidirect product of SO(n) and the little ndiscs operad. Our equivariant recognition principle says that a grouplike space acted on by fDn is equivalent to the nfold loop space on a SO(n)space. Examples of fD2spaces are nerves of ribbon braided monoid ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
The framed ndiscs operad fDn is studied as semidirect product of SO(n) and the little ndiscs operad. Our equivariant recognition principle says that a grouplike space acted on by fDn is equivalent to the nfold loop space on a SO(n)space. Examples of fD2spaces are nerves of ribbon braided monoidal categories. We compute the rational homology of fDn. Koszul duality for semidirect product operads of chain complexes is defined and applied to compute the double loop space homology as BValgebra. MSC(2000): 55P48, 18D10.
OPERADS AND PROPS
, 2006
"... We review definitions and basic properties of operads, PROPs and algebras over these structures. ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
We review definitions and basic properties of operads, PROPs and algebras over these structures.
A Hopf operad of forests of binary trees and related finitedimensional algebras
, 2002
"... The structure of a Hopf operad is defined on the vector spaces spanned by forests of leaflabeled, rooted, binary trees. An explicit formula for the coproduct and its dual product is given, using a poset on forests. 0 ..."
Abstract

Cited by 6 (6 self)
 Add to MetaCart
The structure of a Hopf operad is defined on the vector spaces spanned by forests of leaflabeled, rooted, binary trees. An explicit formula for the coproduct and its dual product is given, using a poset on forests. 0
Homotopy Batalin–Vilkovisky algebras
"... This paper provides an explicit cofibrant resolution of the operad encoding BatalinVilkovisky algebras. Thus it defines the notion of homotopy BatalinVilkovisky algebras with the required homotopy properties. To define this resolution we extend the theory of Koszul duality to operads and properads ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
This paper provides an explicit cofibrant resolution of the operad encoding BatalinVilkovisky algebras. Thus it defines the notion of homotopy BatalinVilkovisky algebras with the required homotopy properties. To define this resolution we extend the theory of Koszul duality to operads and properads that are defined by quadratic and linear relations. The operad encoding Batalin– Vilkovisky algebras is shown to be Koszul in this sense. This allows us to prove a PoincaréBirkhoffWitt Theorem for such an operad and to give an explicit small quasifree resolution for it. This particular resolution enables us to describe the deformation theory and homotopy theory of BValgebras and of homotopy BValgebras. We show that any topological conformal field theory carries a homotopy BValgebra structure which lifts the BValgebra structure on homology. The same result is proved for the singular chain complex of the double loop space of a topological space endowed with an action of the circle. We also prove the cyclic Deligne conjecture with this cofibrant resolution of the operad BV. We develop the general obstruction theory for algebras over the Koszul resolution of a properad and apply it to extend a conjecture of Lian–Zuckerman, showing that certain vertex algebras have an explicit homotopy BValgebra structure.