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22
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.
A resolution (minimal model) of the PROP for bialgebras, preprint math.AT/0209007
"... Abstract. This paper is concerned with a minimal resolution of the prop for bialgebras (Hopf algebras without unit, counit and antipode). We prove a theorem about the form of this resolution (Theorem 12) and give, in Section 5, a lot of explicit formulas for the differential. Our minimal model conta ..."
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Cited by 17 (3 self)
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Abstract. This paper is concerned with a minimal resolution of the prop for bialgebras (Hopf algebras without unit, counit and antipode). We prove a theorem about the form of this resolution (Theorem 12) and give, in Section 5, a lot of explicit formulas for the differential. Our minimal model contains all information about the deformation theory of bialgebras and related cohomology. Algebras over this minimal model are strongly homotopy bialgebras, that is, homotopy invariant versions of bialgebras.
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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Cited by 9 (0 self)
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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.
OPERADS AND PROPS
, 2006
"... We review definitions and basic properties of operads, PROPs and algebras over these structures. ..."
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Cited by 8 (0 self)
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We review definitions and basic properties of operads, PROPs and algebras over these structures.
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 6 (3 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, 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.
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 3 (1 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
Infinite magmatic bialgebras
 Adv. Appl. Math. (2007
"... Abstract. An infinite magmatic bialgebra is a vector space endowed with an nary operation, and an nary cooperation, for each n, verifying some compatibility relations. We prove a rigidity theorem, analogue to the HopfBorel theorem for commutative bialgebras: any connected infinite magmatic bialge ..."
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Cited by 2 (1 self)
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Abstract. An infinite magmatic bialgebra is a vector space endowed with an nary operation, and an nary cooperation, for each n, verifying some compatibility relations. We prove a rigidity theorem, analogue to the HopfBorel theorem for commutative bialgebras: any connected infinite magmatic bialgebra is of the form Mag ∞ (Prim H), where Mag ∞ (V) is the free infinite magmatic algebra over the vector space V. 1.
STRONG HOMOTOPY PROPERADS
, 2007
"... Abstract. In this paper, we define the notion of strong homotopy properads and prove that this structure transfers over left homotopy inverses. We give explicit formulae for the induced structure. 1. ..."
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Abstract. In this paper, we define the notion of strong homotopy properads and prove that this structure transfers over left homotopy inverses. We give explicit formulae for the induced structure. 1.