A Koszul duality for props

Vallette, Bruno

doi:10.1090/s0002-9947-07-04182-7

Cited by 140 publications

(233 citation statements)

References 27 publications

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“…By Theorem 2.6.1, the operad Ᏼ admits a quadratic minimal model. The claim then follows from a straightforward analogue of [Merkulov and Vallette 2007, Theorem 34] (see also [Vallette 2007]) for coloured operads.…”

Section: Definitionmentioning

confidence: 88%

Operad of formal homogeneous spaces and Bernoulli numbers

Merkulov

2008

ANT

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It is shown that for any morphism, φ : g → h, of Lie algebras the vector space underlying the Lie algebra h is canonically a g-homogeneous formal manifold with the action of g being highly nonlinear and twisted by Bernoulli numbers. This fact is obtained from a study of the 2-coloured operad of formal homogeneous spaces whose minimal resolution gives a new conceptual explanation of both Ziv Ran's Jacobi-Bernoulli complex and Fiorenza-Manetti's L∞algebra structure on the mapping cone of a morphism of two Lie algebras. All these constructions are iteratively extended to the case of a morphism of arbitrary L∞-algebras.

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Section: Definitionmentioning

confidence: 88%

Operad of formal homogeneous spaces and Bernoulli numbers

Merkulov

2008

ANT

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“…It should be possible to define a twisted A 1 algebra using the homotopy version of an open Frobenius algebra. The Koszul Duality theory for dioperads described by Gan [9] and for properads described by Vallette [22] provide a definition for such an object. Theorem 3.14 said that the L 1 algebra structure on C ˝H restricts to C ˝Prim.H /.…”

Section: Definition 321mentioning

confidence: 99%

Homotopy algebra structures on twisted tensor products and string topology operations

Miller

2011

Algebr. Geom. Topol.

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Given a C∞ coalgebra C * , a strict dg Hopf algebra H * , and a twisting cochain τ : C * → H * such that Im(τ ) ⊂ P rim(H * ), we describe a procedure for obtaining an A∞ coalgebra on C * ⊗ H * . This is an extension of Brown's work on twisted tensor products. We apply this procedure to obtain an A∞ coalgebra model of the chains on the free loop space LM based on the C∞ coalgebra structure of H * (M ) induced by the diagonal map M → M × M and the Hopf algebra model of the based loop space given by T (H * (M )[−1]). When C * has cyclic C∞ coalgebra structure, we describe an A∞ algebra on C * ⊗ H * . This is used to give an explicit (non-minimal) A∞ algebra model of the string topology loop product. Finally, we discuss a representation of the loop product in principal G-bundles.

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“…This section contains no new material; we merely wish to express the results we need from [9], [7], [8] and [24] in the unifying language of [18].…”

Section: Resolutions Via Koszul Dualitymentioning

confidence: 99%

“…See e.g. [9] for a treatment of quadratic operads, [8] for quadratic dioperads and [24] for quadratic properads and props. We call a (weight graded) G -(co)algebra connected if the underlying S-bimodule is connected.…”

Section: The Decomposition Coproduct Is Defined As Follows For a Decmentioning

confidence: 99%

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Prop profile of bi-Hamiltonian structures

Strohmayer¹

2010

J. Noncommut. Geom.

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Abstract. Recently S. A. Merkulov established a link between differential geometry and homological algebra by giving descriptions of several differential geometric structures in terms of minimal resolutions of props. In particular he described the prop profile of Poisson geometry. In this paper we define a prop such that representations of its minimal resolution in a vector space V are in a one-to-one correspondence with bi-Hamiltonian structures, i.e., pairs of compatible Poisson structures, on the formal manifold associated to V .

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A Koszul duality for props

Abstract: 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.

Cited by 140 publications

References 27 publications

Operad of formal homogeneous spaces and Bernoulli numbers

Operad of formal homogeneous spaces and Bernoulli numbers

Homotopy algebra structures on twisted tensor products and string topology operations

Prop profile of bi-Hamiltonian structures

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