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Gröbner bases for operads
 Duke Math. J
"... Abstract. We define a new monoidal category on collections (shuffle ..."
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Abstract. We define a new monoidal category on collections (shuffle
Iterated bar complexes of Einfinity algebras and homology theories
, 2008
"... We proved in a previous article that the bar complex of an E ∞algebra inherits a natural E ∞algebra structure. As a consequence, a welldefined iterated bar construction B n (A) can be associated to any algebra over an E ∞operad. In the case of a commutative algebra A, our iterated bar constructi ..."
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We proved in a previous article that the bar complex of an E ∞algebra inherits a natural E ∞algebra structure. As a consequence, a welldefined iterated bar construction B n (A) can be associated to any algebra over an E ∞operad. In the case of a commutative algebra A, our iterated bar construction reduces to the standard iterated bar complex of A. The first purpose of this paper is to give a direct effective definition of the iterated bar complexes of E ∞algebras. We use this effective definition to prove that the nfold bar construction admits an extension to categories of algebras over Enoperads. Then we prove that the nfold bar complex determines the homology theory associated to the category of algebras over an Enoperad. In the case n = ∞, we obtain an isomorphism between the homology of an infinite bar construction and the usual Γhomology with trivial coefficients.
PROP profile of deformation quantization
, 2004
"... Using language of dg PROPs we give a new proof of existence of star products on (formal) germs of Poisson manifolds. 1.1. Theorem on quantization of Poisson structures is one of the culminating points of the deformation quantization programme initiated by F. Bayen, M. Flato, C. Fronsdal, A. Lichnero ..."
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Using language of dg PROPs we give a new proof of existence of star products on (formal) germs of Poisson manifolds. 1.1. Theorem on quantization of Poisson structures is one of the culminating points of the deformation quantization programme initiated by F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowics, and D. Sternheimer [BFFLS]. It was first established by Kontsevich in the transcendental work [K1] as a corollary to his formality theorem. Another proof of the formality theorem was
QUADRATIC ALGEBRAS RELATED TO THE BIHAMILTONIAN OPERAD
, 2006
"... Abstract. We prove the conjectures on dimensions and characters of some quadratic algebras stated by B.L.Feigin. It turns out that these algebras are naturally isomorphic to the duals of the components of the bihamiltonian operad. 1. Introduction. All algebras and operads in this paper are defined o ..."
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Abstract. We prove the conjectures on dimensions and characters of some quadratic algebras stated by B.L.Feigin. It turns out that these algebras are naturally isomorphic to the duals of the components of the bihamiltonian operad. 1. Introduction. All algebras and operads in this paper are defined over the field of rational numbers Q. 1.1. Summary of the results. The two following series of quadratic algebras
On a Hopf operad containing the Poisson operad
, 2002
"... Abstract A new Hopf operad Ram is introduced, which contains both the wellknown Poisson operad and the Bessel operad introduced previously by the author. Besides, a structure of cooperad R is introduced on a collection of algebras given by generators and relations which have some similarity with th ..."
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Abstract A new Hopf operad Ram is introduced, which contains both the wellknown Poisson operad and the Bessel operad introduced previously by the author. Besides, a structure of cooperad R is introduced on a collection of algebras given by generators and relations which have some similarity with the Arnold relations for the cohomology of the type A hyperplane arrangement. A map from the operad Ram to the dual operad of R is defined which we conjecture to be a isomorphism. AMS Classification 18D50; 16W30
GENERAL TWISTING OF ALGEBRAS
, 2006
"... ABSTRACT. We introduce the concept of pseudotwistor (with particular cases called twistor and braided twistor) for an algebra (A, µ, u) in a monoidal category, as a morphism T: A ⊗ A → A ⊗ A satisfying a list of axioms ensuring that (A,µ ◦ T, u) is also an algebra in the category. This concept provi ..."
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ABSTRACT. We introduce the concept of pseudotwistor (with particular cases called twistor and braided twistor) for an algebra (A, µ, u) in a monoidal category, as a morphism T: A ⊗ A → A ⊗ A satisfying a list of axioms ensuring that (A,µ ◦ T, u) is also an algebra in the category. This concept provides a unifying framework for various deformed (or twisted) algebras from the literature, such as twisted tensor products of algebras, twisted bialgebras and algebras endowed with Fedosov products. Pseudotwistors appear also in other topics from the literature, e.g. Durdevich’s braided quantum groups and ribbon algebras. We also focus on the effect of twistors on the universal first order differential calculus, as well as on lifting twistors to braided twistors on the algebra of universal differential forms. 1.
An operadic approach to deformation quantization of compatible Poisson brackets
, 2008
"... An analogue of the Livernet–Loday operad for two compatible brackets, which is a flat deformation of the biHamiltonian operad is constructed. The Livernet–Loday operad can be used to define ⋆products and deformation quantization for Poisson structures. The constructed operad is used in the same wa ..."
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An analogue of the Livernet–Loday operad for two compatible brackets, which is a flat deformation of the biHamiltonian operad is constructed. The Livernet–Loday operad can be used to define ⋆products and deformation quantization for Poisson structures. The constructed operad is used in the same way, introducing a definition of operadic deformation quantization of compatible Poisson structures. MSC 2000: 53D55, 18D50 1
Operad profiles of Nijenhuis structures
, 2008
"... Abstract. Recently S. Merkulov [Mer04, Mer05, Mer06] established a new link between differential geometry and homological algebra by giving descriptions of several differential geometric structures in terms of algebraic operads and props. In particular he described Nijenhuis structures as correspond ..."
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Abstract. Recently S. Merkulov [Mer04, Mer05, Mer06] established a new link between differential geometry and homological algebra by giving descriptions of several differential geometric structures in terms of algebraic operads and props. In particular he described Nijenhuis structures as corresponding to representations of the cobar construction on the Koszul dual of a certain quadratic operad. In this paper we prove, using the PBWbasis method of E. Hoffbeck [Hof08], that the operad governing Nijenhuis structures is Koszul, thereby showing that Nijenhuis structures correspond to representations of the minimal resolution of this operad. We also construct an operad such that representations of its minimal resolution in a vector space V are in onetoone correspondence with pairs of compatible Nijenhuis structures on the formal manifold associated to V.
KOSZUL DUALITY OF EnOPERADS
"... Abstract. The goal of this paper is to prove a Koszul duality result for Enoperads in differential graded modules over a ring. The case of an E1operad, which is equivalent to the associative operad, is classical. For n> 1, the homology of an Enoperad is identified with the nGerstenhaber operad an ..."
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Abstract. The goal of this paper is to prove a Koszul duality result for Enoperads in differential graded modules over a ring. The case of an E1operad, which is equivalent to the associative operad, is classical. For n> 1, the homology of an Enoperad is identified with the nGerstenhaber operad and forms another well known Koszul operad. Our main theorem asserts that an operadic cobar construction on the dual cooperad of an Enoperad En defines a cofibrant model of En. This cofibrant model gives a realization at the chain level of the minimal model of the nGerstenhaber operad arising from Koszul duality. Most models of Enoperads in differential graded modules come in nested sequences E1 ⊂ E2 ⊂ · · · ⊂ E ∞ homotopically equivalent to the sequence of the chain operads of little cubes. In our main theorem, we also define a model of the operad embeddings En−1 ↩ → En at the level of cobar constructions.
Iterated bar complexes of E∞ algebras and homology theories
, 2010
"... We proved in a previous article that the bar complex of an E∞algebra inherits a natural E∞algebra structure. As a consequence, a welldefined iterated bar construction Bn (A) can be associated to any algebra over an E∞operad. In the case of a commutative algebra A, our iterated bar construction re ..."
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We proved in a previous article that the bar complex of an E∞algebra inherits a natural E∞algebra structure. As a consequence, a welldefined iterated bar construction Bn (A) can be associated to any algebra over an E∞operad. In the case of a commutative algebra A, our iterated bar construction reduces to the standard iterated bar complex of A. The first purpose of this paper is to give a direct effective definition of the iterated bar complexes of E∞algebras. We use this effective definition to prove that the nfold bar construction admits an extension to categories of algebras over Enoperads. Then we prove that the nfold bar complex determines the homology theory associated to the category of algebras over an Enoperad. In the case n = ∞, we obtain an isomorphism between the homology of an infinite bar construction and the usual Γhomology with trivial coefficients.