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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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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 19 (5 self)
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We review definitions and basic properties of operads, PROPs and algebras over these structures.
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 18 (6 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.
Nijenhuis infinity and contractible dg manifolds
 Arxiv math.AG/0403244, Compositio Mathematica
"... We find a minimal differential graded (dg) operad whose generic representations in R n are in onetoone correspondence with formal germs of those endomorphisms of the tangent bundle to R n which satisfy the Nijenhuis integrability condition. This operad is of a surprisingly simple origin — it is th ..."
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We find a minimal differential graded (dg) operad whose generic representations in R n are in onetoone correspondence with formal germs of those endomorphisms of the tangent bundle to R n which satisfy the Nijenhuis integrability condition. This operad is of a surprisingly simple origin — it is the cobar construction on the quadratic operad of homologically trivial dg Lie algebras. As a by product we obtain a strong homotopy generalization of this geometric structure and show its homotopy equivalence to the structure of contractible dg manifold. 1. Introduction. Extended
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 6 (4 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.
Symmetric Boolean algebras
"... We define Boolean algebras in the linear context and study its symmetric powers. We give explicit formulae for products in symmetric Boolean algebras of various dimensions. We formulate symmetric forms of the inclusionexclusion principle. ..."
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We define Boolean algebras in the linear context and study its symmetric powers. We give explicit formulae for products in symmetric Boolean algebras of various dimensions. We formulate symmetric forms of the inclusionexclusion principle.
DEFORMATION THEORY OF REPRESENTATIONS OF PROP(ERAD)S II
"... Abstract. This paper is the followup of [MV08]. ..."
STRONGLY HOMOTOPY LIE BIALGEBRAS AND LIE QUASIBIALGEBRAS
, 2007
"... This paper is dedicated to JeanLouis Loday on the occasion of his 60th birthday with admiration and gratitude Structures of Lie algebras, Lie coalgebras, Lie bialgebras and Lie quasibialgebras are presented as solutions of MaurerCartan equations on corresponding governing differential graded Lie a ..."
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This paper is dedicated to JeanLouis Loday on the occasion of his 60th birthday with admiration and gratitude Structures of Lie algebras, Lie coalgebras, Lie bialgebras and Lie quasibialgebras are presented as solutions of MaurerCartan equations on corresponding governing differential graded Lie algebras using the big bracket construction of KosmannSchwarzbach. This approach provides a definition of an L∞(quasi)bialgebra (strongly homotopy Lie (quasi)bialgebra). We recover an L∞algebra structure as a particular case of our construction. The formal geometry interpretation leads to a definition of an L ∞ (quasi)bialgebra structure on V as a differential operator Q on V, selfcommuting with respect to the big bracket. Finally, we establish an L∞version of a Manin (quasi) triple and get a correspondence theorem with L∞(quasi) bialgebras. 1. Introduction. Algebraic structures are often defined as certain maps which must satisfy quadratic relations. One of the examples is a Lie algebra structure: a Lie bracket satisfies the Jacobi identity (indeed the Jacobi identity is a quadratic relation since the bracket appears twice in each summand). Other examples include an associative multiplication (the associativity condition is quadratic), L ∞ and
AN UPDATE ON SEMISIMPLE QUANTUM COHOMOLOGY AND F–MANIFOLDS
, 803
"... Abstract. In the first section of this note we show that the Theorem 1.8.1 of Bayer–Manin ([BaMa]) can be strengthened in the following way: if the even quantum cohomology of a projective algebraic manifold V is generically semi–simple, then V has no odd cohomology and is of Hodge–Tate type. In part ..."
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Abstract. In the first section of this note we show that the Theorem 1.8.1 of Bayer–Manin ([BaMa]) can be strengthened in the following way: if the even quantum cohomology of a projective algebraic manifold V is generically semi–simple, then V has no odd cohomology and is of Hodge–Tate type. In particular, this addressess a question in [Ci]. In the second section, we prove that an analytic (or formal) supermanifold M with a given supercommutative associative OM–bilinear multiplication on its tangent sheaf TM is an F–manifold in the sense of [HeMa], iff its spectral cover as an analytic subspace of the cotangent bundle T ∗ M is coisotropic of maximal dimension. This answers a question of V. Ginzburg. Finally, we discuss these results in the context of mirror symmetry and Landau– Ginzburg models for Fano varieties.
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.