Results 1  10
of
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. ..."
Abstract

Cited by 21 (4 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 12 (5 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 9 (4 self)
 Add to MetaCart
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
, 2007
"... 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 provid ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
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.
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. ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
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
INTERNAL COHOMOMORPHISMS FOR OPERADS
, 2007
"... 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 ” structur ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
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.
5.2. Dupont’s chain homotopy 75
"... 4.4. Effective action for BF theory as a generating function for algebraic structure on the ..."
Abstract
 Add to MetaCart
4.4. Effective action for BF theory as a generating function for algebraic structure on the
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 ..."
Abstract
 Add to MetaCart
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 A ss as rather nonobvious extensions of Com ∞ and A ss∞, involving, e.g., a mysterious mixture of associahedra with cyclohedra. Finally, we apply the above results to a computation of