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13
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.
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.
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.
Generalized operads and their inner cohomomorphisms, arXiv:math.CT/ 0609748
, 2006
"... Abstract. In this paper we introduce a notion of generalized operad containing as special cases various kinds of operad–like objects: ordinary, cyclic, modular, properads etc. We then construct inner cohomomorphism objects in their categories (and categories of algebras over them). We argue that the ..."
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Cited by 8 (1 self)
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Abstract. In this paper we introduce a notion of generalized operad containing as special cases various kinds of operad–like objects: ordinary, cyclic, modular, properads etc. We then construct inner cohomomorphism objects in their categories (and categories of algebras over them). We argue that they provide an approach to symmetry and moduli objects in noncommutative geometries based upon these “ring–like ” structures. We give a unified axiomatic treatment of generalized operads as functors on categories of abstract labeled graphs. Finally, we extend inner cohomomorphism constructions to more general categorical contexts. This version differs from the previous ones by several local changes (including the title) and two extra references. 0.1. Inner cohomomorphisms of associative algebras. Let k be a field. Consider pairs A = (A, A1) consisting of an associative k–algebra A and a finite dimensional subspace A1 generating A. For two such pairs A = (A, A1) and B =
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.
Differential operators and BV structures in noncommutative geometry
, 710
"... We introduce a new formalism of differential operators for a general associative algebra A. It replaces Grothendieck’s notion of differential operator on a commutative algebra in such a way that derivations of the commutative algebra are replaced by DerA, the bimodule of double derivations. Our diff ..."
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Cited by 2 (0 self)
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We introduce a new formalism of differential operators for a general associative algebra A. It replaces Grothendieck’s notion of differential operator on a commutative algebra in such a way that derivations of the commutative algebra are replaced by DerA, the bimodule of double derivations. Our differential operators act not on the algebra A itself but rather on F(A), a certain ‘Fock space ’ associated to any noncommutative algebra A in a functorial way. The corresponding algebra D(F(A)), of differential operators, is filtered and gr D(F(A)), the associated graded algebra, is commutative in some ‘twisted ’ sense. The resulting double Poisson structure on gr D(F(A)) is closely related to the one introduced by Van den Bergh. Specifically, we prove that gr D(F(A)) ∼ = F(TA(DerA)), provided the algebra A is smooth. It is crucial for our construction that the Fock space F(A) carries an extrastructure of a wheelgebra, a new notion closely related to the notion of a wheeled PROP. There are also notions of Lie wheelgebras, and so on. In that language, D(F(A)) becomes the universal enveloping wheelgebra of a Lie wheelgebroid of double derivations. In the second part of the paper we show, extending a classical construction of Koszul to the
Deformation Quantization and Reduction
, 2007
"... Abstract. This note is an overview of the Poisson sigma model (PSM) and its applications in deformation quantization. Reduction of coisotropic and prePoisson submanifolds, their appearance as branes of the PSM, quantization in terms of L∞ and A∞algebras, and bimodule structures are recalled. As a ..."
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Cited by 1 (1 self)
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Abstract. This note is an overview of the Poisson sigma model (PSM) and its applications in deformation quantization. Reduction of coisotropic and prePoisson submanifolds, their appearance as branes of the PSM, quantization in terms of L∞ and A∞algebras, and bimodule structures are recalled. As an application, an “almost ” functorial quantization of Poisson maps is presented if no anomalies occur. This leads in principle to a novel approach for the quantization of Poisson–Lie groups. 1.
Contents
, 2007
"... Abstract. 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 ..."
Abstract
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Abstract. 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 Ass as rather nonobvious extensions of Com ∞ and Ass∞, involving, e.g., a mysterious mixture of associahedra with cyclohedra. Finally, we apply the above results to a computation of
INTERNAL COHOMOMORPHISMS FOR OPERADS 1
, 2007
"... Abstract. In this paper we construct internal cohomomorphism objects in various categories of operads (ordinary, cyclic, modular, properads...) and algebras over operads. We argue that they provide an approach to symmetry and moduli objects in noncommutative geometries based upon these “ring–like ” ..."
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Abstract. In this paper we construct internal cohomomorphism objects in various categories of operads (ordinary, cyclic, modular, properads...) and algebras over operads. We argue that they provide an approach to symmetry and moduli objects in noncommutative geometries based upon these “ring–like ” structures. We give also a unified axiomatic treatment of operads as functors on labeled graphs. Finally, we extend internal cohomomorphism constructions to more general categorical contexts. 0.1. Internal cohomomorphisms of associative algebras. Let k be a field. Consider pairs A = (A, A1) consisting of an associative k–algebra A and a finite dimensional subspace A1 generating A. For two such pairs A = (A, A1) and
CHARACTERISTIC CLASSES OF QMANIFOLDS: CLASSIFICATION AND APPLICATIONS
, 906
"... Abstract. A Qmanifold M is a supermanifold endowed with an odd vector field Q squaring to zero. The Lie derivative LQ along Q makes the algebra of smooth tensor fields on M into a differential algebra. In this paper, we define and study the invariants of Qmanifolds called characteristic classes. T ..."
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Abstract. A Qmanifold M is a supermanifold endowed with an odd vector field Q squaring to zero. The Lie derivative LQ along Q makes the algebra of smooth tensor fields on M into a differential algebra. In this paper, we define and study the invariants of Qmanifolds called characteristic classes. These take values in the cohomology of the operator LQ and, given an affine symmetric connection with curvature R, can be represented by universal tensor polynomials in the repeated covariant derivatives of Q and R up to some finite order. As usual, the characteristic classes are proved to be independent of the choice of the affine connection used to define them. The main result of the paper is a complete classification of the intrinsic characteristic classes, which, by definition, do not vanish identically on flat Qmanifolds. As an illustration of the general theory we interpret some of the intrinsic characteristic classes as anomalies in the BV and BFVBRST quantization methods of gauge theories. An application to the theory of (singular) foliations is also discussed. 1.