A KOSZUL DUALITY FOR PROPS

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A KOSZUL DUALITY FOR PROPS BRUNO VALLETTE 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. Introduction The Koszul duality is a theory developed for the first time in 1970 by S. Priddy for associative algebras in [Pr]. To every quadratic algebra A, it associates a dual coal- gebra A¡ and a chain complex called Koszul complex. When this complex is acyclic, we say that A is a Koszul algebra. In this case, the algebra A and its represen- tations have many properties (cf. A. Beilinson, V. Ginzburg and W. Soergel [BGS]). In 1994, this theory was generalized to algebraic operads by V. Ginzburg and M.M. Kapranov (cf. [GK]). An operad is an algebraic object that models the opera- tions with n inputs (and one output) A?n ? A acting on a type of algebras. For instance, there exists operads As, Com and Lie coding associative, commutative and Lie algebras. The Koszul duality theory for operads has many applications: construction of a “small” chain complex to compute the homology groups of an algebra, minimal model of an operad, notion of algebra up to homotopy.

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Aguiar Maxim Kontsevich

A KOSZUL DUALITY FOR

BRUNO VALLETTE

PROPS

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.

Introduction

The Koszul duality is a theory developed for the ﬁrst time in 1970 by S. Priddy for associative algebras in [Pr]. To every quadratic algebraA, it associates a dual coal-gebraA¡ this complex is acyclic, Whenand a chain complex called Koszul complex. we say thatAis a Koszul algebra. In this case, the algebraAand its represen-tations have many properties (cf.A. Beilinson, V. Ginzburg and W. Soergel [BGS]).

In 1994, this theory was generalized to algebraic operads by V. Ginzburg and M.M. Kapranov (cf.[GK]). An operad is an algebraic object that models the opera-tions withninputs (and one output)A⊗n→Aacting on a type of algebras. For instance, there exists operadsAs,ComandLiecoding associative, commutative and Lie algebras. The Koszul duality theory for operads has many applications: construction of a “small” chain complex to compute the homology groups of an algebra, minimal model of an operad, notion of algebra up to homotopy.

The discovery of quantum groups (cf.V. Drinfeld [Dr1, Dr2]) has popularized algebraic structures with products and coproducts, that’s-to-say operations with multiple inputs and multiple outputsA⊗n→A⊗m is the case of bialgebras,. It Hopf algebras and Lie bialgebras, for instance. The framework of operads is too narrow to treat such structures. To model the operations with multiple inputs and outputs, one has to use a more general algebraic object : the props. Following J.-P. Serre in [S], we call an “algebra” over a propP, aP-gebra.

It is natural to try to generalize Koszul duality theory to props. Few works has been done in that direction by M. Markl and A.A. Voronov in [MV] and W.L. Gan in [G]. Actually, M. Markl and A.A. Voronov proved Koszul duality theory for what M. Kontsevich calls21-PROPs and W.L. Gan proved it for dioperads. One has to bear in mind that an associative algebra is an operad, an operad is a12-PROP, a 21-PROP is a dioperad and a dioperad induces a prop.

2000Mathematics Subject Classiﬁcation.18D50 (16W30, 17B26, 55P48). Keywords and phrases.Prop, Koszul Duality, Operad, Lie Bialgebra, Frobenius Algebra.

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