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16
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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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.
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
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.
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
Quantization of strongly homotopy Lie bialgebras, ArXiv Mathematics eprints
, 2006
"... Abstract. Using theory of props we prove a formality theorem associated with universal quantizations of (strongly homotopy) Lie bialgebras. 1. ..."
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Cited by 4 (2 self)
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Abstract. Using theory of props we prove a formality theorem associated with universal quantizations of (strongly homotopy) Lie bialgebras. 1.
A concept of 2 3PROP and deformation theory of (co)associative coalgebras
, 2003
"... To my teacher Borya Feigin in the occasion of his 50th birthday We introduce a concept of 2 3PROP generalizing the Kontsevich concept of 1 2PROP. We prove that some Stashefftype compactification of the Kontsevich spaces K(m, n) defines a topological 2 3PROP structure. The corresponding chain comple ..."
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Cited by 3 (0 self)
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To my teacher Borya Feigin in the occasion of his 50th birthday We introduce a concept of 2 3PROP generalizing the Kontsevich concept of 1 2PROP. We prove that some Stashefftype compactification of the Kontsevich spaces K(m, n) defines a topological 2 3PROP structure. The corresponding chain complex is a minimal model for its cohomology (both are considered as 2 3PROPs). We construct a 2PROP End(V) for a vector space V. Finally, we construct a
D.: The L∞deformation complex of diagrams of algebras
 Cambridge Studies in Advanced Mathematics
, 1994
"... Abstract. The deformation complex of an algebra over a colored PROP P is defined in terms of a minimal (or, more generally, cofibrant) model of P. It is shown that it carries the structure of an L∞algebra which induces a graded Lie bracket on cohomology. As an example, the L∞algebra structure on t ..."
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Abstract. The deformation complex of an algebra over a colored PROP P is defined in terms of a minimal (or, more generally, cofibrant) model of P. It is shown that it carries the structure of an L∞algebra which induces a graded Lie bracket on cohomology. As an example, the L∞algebra structure on the deformation complex of an associative algebra morphism g is constructed. Another example is the deformation complex of a Lie algebra morphism. The last example is the diagram describing two mutually inverse morphisms of vector spaces. Its L∞deformation complex has nontrivial l0term. Explicit formulas for the L∞operations in the above examples are given. A typical deformation complex of a diagram of algebras is a fullyfledged L∞algebra with nontrivial higher operations. Contents
Contents
"... Abstract. We give an explicit construction of the free monoid in monoidal abelian categories when the monoidal product does not necessarily preserve coproducts. Then we apply it to several new monoidal categories that appeared recently in the theory of Koszul duality for operads and props. This give ..."
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Abstract. We give an explicit construction of the free monoid in monoidal abelian categories when the monoidal product does not necessarily preserve coproducts. Then we apply it to several new monoidal categories that appeared recently in the theory of Koszul duality for operads and props. This gives a conceptual explanation of the form of the free operad, free dioperad and free
DEFORMATION THEORY OF REPRESENTATIONS OF PROP(ERAD)S II
"... Abstract. This paper is the followup of [MV08]. ..."