Results 1  10
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15
A Koszul duality for props
 Trans. of Amer. Math. Soc
"... Abstract. The notion of prop models the operations with multiple inputs and multiple outputs, acting on some algebraic structures like the bialgebras or the Lie bialgebras. In this paper, we generalize the Koszul duality theory of associative algebras and operads to props. ..."
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Cited by 21 (4 self)
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Abstract. The notion of prop models the operations with multiple inputs and multiple outputs, acting on some algebraic structures like the bialgebras or the Lie bialgebras. In this paper, we generalize the Koszul duality theory of associative algebras and operads to props.
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 19 (4 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.
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 ..."
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Cited by 15 (4 self)
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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
WHEELED PROPS, GRAPH COMPLEXES AND THE MASTER EQUATION
, 2007
"... We introduce and study wheeled PROPs, an extension of the theory of PROPs which can treat traces and, in particular, solutions to the master equations which involve divergence operators. We construct a dg free wheeled PROP whose representations are in onetoone correspondence with formal germs of ..."
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Cited by 12 (5 self)
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We introduce and study wheeled PROPs, an extension of the theory of PROPs which can treat traces and, in particular, solutions to the master equations which involve divergence operators. We construct a dg free wheeled PROP whose representations are in onetoone correspondence with formal germs of SPmanifolds, key geometric objects in the theory of BatalinVilkovisky quantization. We also construct minimal wheeled resolutions of classical operads Com and As s as rather nonobvious extensions of Com ∞ and As s∞, involving, e.g., a mysterious mixture of associahedra with cyclohedra. Finally, we apply the above results to a computation of cohomology of a directed version of Kontsevich’s complex of ribbon graphs.
Profile of Poisson Geometry
 Comm. Math.Phys
"... “The genetic code appears to be universal;... ” ..."
DEFORMATION THEORY OF REPRESENTATIONS OF PROP(ERAD)S I
"... Abstract. In this paper and its followup [MV08], we study the deformation theory of morphisms of properads and props thereby extending Quillen’s deformation theory for commutative rings to a nonlinear framework. The associated chain complex is endowed with an L∞algebra structure. Its MaurerCarta ..."
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Cited by 9 (4 self)
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Abstract. In this paper and its followup [MV08], we study the deformation theory of morphisms of properads and props thereby extending Quillen’s deformation theory for commutative rings to a nonlinear framework. The associated chain complex is endowed with an L∞algebra structure. Its MaurerCartan elements correspond to deformed structures, which allows us to give a geometric interpretation of these results.
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.
Gröbner bases for operads
 Duke Math. J
"... Abstract. We define a new monoidal category on collections (shuffle ..."
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Cited by 6 (2 self)
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Abstract. We define a new monoidal category on collections (shuffle
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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Cited by 4 (0 self)
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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
NATURAL DIFFERENTIAL OPERATORS AND GRAPH COMPLEXES
, 2007
"... We show how the machine invented by S. Merkulov [19, 20, 22] can be used to study and classify natural operators in differential geometry. We also give an interpretation of graph complexes arising in this context in terms of representation theory. As application, we prove several results on classifi ..."
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Cited by 4 (3 self)
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We show how the machine invented by S. Merkulov [19, 20, 22] can be used to study and classify natural operators in differential geometry. We also give an interpretation of graph complexes arising in this context in terms of representation theory. As application, we prove several results on classification of natural operators acting on vector fields and connections.