Abstract: Shuffles operads are an intermediate notion between non-symmetric operads and symmetric operads. They provide a suitable framework where rewriting theory can be applied, for instance to study homology properties of symmetric operads. We will recall the definition of shuffle operads and the main constructions in this framework. We will then show how operadic Gröbner bases can be used to obtain results for symmetric operads, for instance Koszulness, or more generally to compute Quillen homology.
Abstract: The general study of Koszul resolutions have been made by Priddy to simplify the computations of the cohomology groups of associative algebras. It applies to algebras with a nice presentation. The simplest setting is given by associative algebras having a quadratic presentation. Nevertheless, the Koszul duality extends in different manner to inhomogeneous algebras. We will present an overview of the possible generalizations. We will be also interested in extending the quadratic setting to homogeneous relations of the same degree N possibly greater than 2. Two main examples of N-Koszul algebras will be studied: the N-symmetric algebras (which are related to the Mac Mahon Master Theorem), and the algebras defined by one relation of degree N (which are related to the Gerasimov theorem).
Abstract: Inversion of formal power series is closely related to the theory of Koszul dual operads. In the particular case of an algebraic generating series F(x), the inverse is very easy to deal with, as it satisfy just the same equation. I will present a simple algorithm to compute a slightly more refined algebraic equation involving one more parameter and describing the bar complex. The proof makes use of tree-indexed series.
Abstract: We'll recall the notions of (N-homogeneous) X-confluent algebra and confluence algebra. Using the confluence algebra, Roland Berger constructed a contracting homotopy for the Koszul complex of any quadratic X-confluent algebra. He deduced that such algebras are Koszul. We'll see that it is possible to construct this contracting homotopy studying the representations of the confluence algebra. Moreover, we'll also study the case of N-homogeneous X-confluent algebras.
Abstract: I shall explain how to define Gröbner bases for algebras over an arbitrary non-symmetric operad. The main observation allowing that is that operads with arity 0 operations do not create rewriting problems in the non-symmetric case. This will be applied to Yoneda algebras of N-homogeneous algebras to characterize N-Koszulness from the A-infinity viewpoint. The talk is based on a joint work with Bruno Vallette.
Abstract:There are two basic approaches to defining
operadic (or multi) and dioperadic (or poly) composition. These differ on whether composition is done
simultaneously or partially. The two notions being equivalent in the presence
of units, but not necessarily otherwise.
Simultaneous operadic and dioperadic composition structures have been studied categorically and exhibited as associative algebra (or semigroup object) structures. In this talk I will focus on the notions of operadic and dioperadic partial composition, relating them to Lie algebra structure. Specifically, I will give combinatorial interpretations for the pre-Lie and the Lie-admissible identities by means of natural isomorphisms in species of structures, and show that their associated algebras respectively characterise symmetric multicategories and symmetric polycategories. The development is part of a wider categorical theory of operads with algebraic structure which, if time allows, I will also touch upon.
Connections with algebraic operads will be drawn and related open problems will be discussed.
Abstract: We introduce and advocate our notion of non-symmetric modular operads. An immediate application of our theory will be a short proof that the modular envelope of the associative operad is the symmetrization of the terminal operad in the category of nonsymmetric modular operads. This gives a succint description of this object that plays the basic role in open string field theory.
Abstract: E_n-algebras, algebraic analogues of n-fold loop spaces, come with a suitable notion of homology, called E_n-homology. E_n-homology can be calculated via a generalized iterated bar construction. In this talk, I will explain how to interpret E_n-cohomology of a commutative algebra with coeffcients in a symmetric bimodule as functor cohomology and discuss the Yoneda pairing in this context.
Pierre-Louis Curien (PPS, Paris 7), Stéphane Gaussent (ICJ, Saint-Étienne), Eric Hoffbeck (LAGA, Paris 13), Philippe Malbos (ICJ, Lyon 1), Joan Millès (IMT, Paul Sabatier), Samuel Mimram (CEA LIST).