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33
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 ..."
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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 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 ..."
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Cited by 35 (4 self)
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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.
OPERADS AND PROPS
, 2006
"... We review definitions and basic properties of operads, PROPs and algebras over these structures. ..."
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Cited by 20 (5 self)
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We review definitions and basic properties of operads, PROPs and algebras over these structures.
MANIN PRODUCTS, KOSZUL DUALITY, LODAY ALGEBRAS AND DELIGNE CONJECTURE
"... Dedicated to JeanLouis Loday, on the occasion of his sixtieth birthday 1 Abstract. In this article we give a conceptual definition of Manin products in any category endowed with two coherent monoidal products. This construction can be applied to associative algebras, nonsymmetric operads, operads, ..."
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Dedicated to JeanLouis Loday, on the occasion of his sixtieth birthday 1 Abstract. In this article we give a conceptual definition of Manin products in any category endowed with two coherent monoidal products. This construction can be applied to associative algebras, nonsymmetric operads, operads, colored operads, and properads presented by generators and relations. These two products, called black and white, are dual to each other under Koszul duality functor. We study their properties and compute several examples of black and white products for operads. These products allow us to define natural operations on the chain complex defining cohomology theories. With these operations, we are able to prove that Deligne’s conjecture holds for a general class of operads and is not specific to the case of associative algebras. Finally, we prove generalized versions of a few conjectures raised by M. Aguiar and J.L. Loday related to the Koszul property of operads defined by black products. These operads provide infinitely many examples for this generalized Deligne’s conjecture.
Graph complexes with loops and wheels
 In Yu Tschinkel and Yu Zarhin, editors, Algebra, Arithmetic and Geometry  Manin Festschrift. Birkhäuser
, 2008
"... 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 [Kon93, Kon02] as a way to expose highly nontrivial interrelati ..."
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Cited by 6 (3 self)
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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 [Kon93, Kon02] as a way to expose highly nontrivial interrelations
Operads, clones, and distributive laws
, 2008
"... Abstract We show how nonsymmetric operads (or multicategories), symmetric operads, and clones, arise from three suitable monads on Cat, each extending to a monad on profunctors thanks to a distributivelaw. The presentation builds upon recent work by Fiore, Gambino, Hyland, and Winskel on a theory ..."
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Abstract We show how nonsymmetric operads (or multicategories), symmetric operads, and clones, arise from three suitable monads on Cat, each extending to a monad on profunctors thanks to a distributivelaw. The presentation builds upon recent work by Fiore, Gambino, Hyland, and Winskel on a theory of generalized species of structures,but, for the multicategory case, the general idea goes back to Burroni's Tcategories (1971). We show how other previous categorical analysesof operad (via Day's tensor products, or via analytical functor) fit with the profunctor approach.
The odd origin of Gerstenhaber, BV and the master equation
, 2012
"... Abstract. In this paper we show that Gerstenhaber brackets, BV operators and related master equations arise in a very natural way when considering odd operads and their generalizations. We show that many known examples such as BV operators in the Calabi– Yau setting, brackets in string field theor ..."
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Abstract. In this paper we show that Gerstenhaber brackets, BV operators and related master equations arise in a very natural way when considering odd operads and their generalizations. We show that many known examples such as BV operators in the Calabi– Yau setting, brackets in string field theory, the master equation in that setting, the master equation for Feynman transforms come from this type of setup. We give a systematic and comprehensive treatment of all the usual setups involving (cyclic/modular) operads and PROP(erad)s including new results. Further generalizations and categorical constructions will be presented in a sequel.
DEFORMATION THEORY OF REPRESENTATIONS OF PROP(ERAD)S II
"... Abstract. This paper is the followup of [MV08]. ..."
STRONGLY HOMOTOPY LIE BIALGEBRAS AND LIE QUASIBIALGEBRAS
, 2007
"... This paper is dedicated to JeanLouis Loday on the occasion of his 60th birthday with admiration and gratitude Structures of Lie algebras, Lie coalgebras, Lie bialgebras and Lie quasibialgebras are presented as solutions of MaurerCartan equations on corresponding governing differential graded Lie a ..."
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This paper is dedicated to JeanLouis Loday on the occasion of his 60th birthday with admiration and gratitude Structures of Lie algebras, Lie coalgebras, Lie bialgebras and Lie quasibialgebras are presented as solutions of MaurerCartan equations on corresponding governing differential graded Lie algebras using the big bracket construction of KosmannSchwarzbach. This approach provides a definition of an L∞(quasi)bialgebra (strongly homotopy Lie (quasi)bialgebra). We recover an L∞algebra structure as a particular case of our construction. The formal geometry interpretation leads to a definition of an L ∞ (quasi)bialgebra structure on V as a differential operator Q on V, selfcommuting with respect to the big bracket. Finally, we establish an L∞version of a Manin (quasi) triple and get a correspondence theorem with L∞(quasi) bialgebras. 1. Introduction. Algebraic structures are often defined as certain maps which must satisfy quadratic relations. One of the examples is a Lie algebra structure: a Lie bracket satisfies the Jacobi identity (indeed the Jacobi identity is a quadratic relation since the bracket appears twice in each summand). Other examples include an associative multiplication (the associativity condition is quadratic), L ∞ and