In this talk, I briefly recall the Loday algebras and their relationships with Lie and pre-Lie algebras, which lead to the introduction of Lie analogues of Loday algebras. These algebraic structures provide examples of successors of operads, which are interpreted in terms of Manin black products. [Slides] [Top]

PBW theorem for Lie algebras has two formulations. The quality one is that any Lie algebra over a field can be embedded into an associative algebra (hence into its universal enveloping algebra). The quantity one gives a linear basis of the universal enveloping algebra of any Lie algebra presented by a multiplication table. More general quantity type formulation of the PBW theorem for Lie algebras is as follows: Let $L=Lie(X|S)$ be a presentation of a Lie algebra over a field with a Gröbner-Shirshov basis $S$. Then $S^{(-)}$, the set of noncommutative polynomials that is the result of working out all Lie brackets $[x,y]=xy-yx$ in polynomials from $S$, is a Gröbner-Shirshov basis of $U(L)$ in the generators $X$. The Cohn-Shirshov theorem on speciality any Jordan algebra with 2 generators is an example of PBW quality type theorem for Jordan algebras (and we don't know a quantity type theorem). I will speak about applications of Gröbner-Shirshov bases to the quantity and quality type PBW theorems for different classes of algebras. For any Lie conformal algebra $L=LieConf (X|S)$, where $X$ is a linear basis of $L$, only ``1/2 PBW theorem" is valid: only some compositions (of intersection) are trivial in the universal enveloping associative conformal algebras $U(L)= AConf (X|S^{(-)})$. [Top]

I will explain how one can use several operad structures to describe nice properties of the classical Tamari posets and possibly also of the very recent Tamari posets of higher slope. [Top]

Gröbner-Shirshov bases theory for metabelian Lie algebras was first considered by V.V. Talapov in 1982. However, there are serious gaps in his paper. He missed several cases when he defined compositions. This means the theory was not established correctly. We refine his idea and complete the results. It is well-known that for many kinds of algebras, for example, associative algebras, Lie algebras, etc., if $A=(X|S_X)$ and $B=(Y|S_Y)$ are defined by generators and relations, where $S_X$ and $S_Y$ are Gröbner-Shirshov bases respectively, then $S_X\cup S_Y$ is a Gröbner-Shirshov basis for the free product $A*B = (X\cup Y | S_X\cup S_Y)$ of $A$ and $B$. We prove that it is not the case for metabelian Lie algebras, even in the case of $S_Y = \emptyset$. As more applications, we find the Gröbner-Shirshov bases for partial commutative metabelian Lie algebras related to circuits, trees and some cubes. [Slides] [Top]

Let $k$ be a field. A $k$-linear space $D$ equipped with two bilinear multiplications $\vdash$ and $\dashv $ is called a dialgebra, if both $\vdash$ and $\dashv$ are associative and \begin{eqnarray*} a\dashv(b\vdash c)&=&a\dashv b\dashv c \\ (a\dashv b)\vdash c&=&a\vdash b\vdash c \\ a\vdash(b\dashv c)&=&(a\vdash b)\dashv c \end{eqnarray*} for any $a, \ b, \ c\in D$. A normal form for free dialgebra is found by J.-L. Loday. By using this result, we establish the Gröbner-Shirshov bases theory for dialgebras. As results, we give Gröbner-Shirshov bases for the universal enveloping algebra of a Leibniz algebra, the bar extension of a dialgebra, the free product of two dialgebras, and Clifford dialgebra. We obtain some normal forms for algebras mentioned the above. [Top]

I shall discuss the notion of a filtered distributive law between two operads, and related open questions. One new series of filtered distributive laws which will be the key example in the talk is of topological origin; quite unexpectedly, it includes a filtered distributive law between "very non-topological" operads Perm and NAP. This talk is based on a joint work with James Griffin. [Slides] [Top]

Lie commutator in n-ary case is defined as a skew-symmetric sum of n! compositions by all permutations. Poisson bracket in n-ary case is defined as a linear combinatiion of jacobians. We find identities for n-ary associative algebras under n-ary Lie commutators and identities for n-ary Jacobian algebras under n-ary Poisson brackets. Questions about such identities was posed by A. G. Kurosh and V. T. Filippov. [Slides] [Top]

I will report on a joint work with Yves Guiraud and Philippe Malbos. For a given presentation of a monoid M, we consider its associated 2-polygraph. Then using a machinery based on higher rewriting theory, we obtain a homotopy basis. This homotopy basis is exactly the piece of information one needs to get the coherence diagrams involved in a (clever) definition of an action of M on a category. In some good cases, it is possible to simplify this basis. Namely, if we start with the Deligne's presentation of the positive braid monoid, we obtain a simplified version of a result of Deligne. [Top]

Rota asked the question of classifying all linear operators that can be defined on associative algebras. There are quite a few such operators, such as endomorphisms, derivations and Rota-Baxter operators. We put the question in the framework of operated algebras and operads, and explore the relationship between the classification and rewriting systems. [Slides] [Top]

In this talk, I will introduce the notion of Poincaré-Birkhoff-Witt bases for N-Koszul algebras. The notion of N-Koszul was introduced by Roland Berger in the early 2000's to generalize the Koszul property, with homogeneous relations in a fixed weight N rather than in weight 2. PBW bases will allow us to prove easily the N-Koszulity of some algebras. They can be seen as Gröbner bases satisfying an extra condition. [Top]

A dialogue category is a monoidal category equipped with an exponentiating object. In this talk, I will introduce a notion of braided dialogue category, and explain how this notion provides a functorial bridge between proof theory and knot theory. [Top]

I will show how presentations of monoids by rewriting systems lead to higher categories, and discuss a Quillen model structure on strict omega-categories from which homological and homotopical invariants of monoids may be recovered. This is joint work with Yves Lafont and Krzysztof Worytkiewicz. [Slides] [Top]

Rewriting systems on words are very useful in the study of monoids. In good cases, they give finite presentations of the monoids, allowing their manipulation by a computer. Even better, when the presentation is confluent and terminating, they provide one with a notion of canonical representative for the elements of the presented monoid. Polygraphs are a higher-dimensional generalization of this notion of presentation, from monoids to n-categories. Here, we are interested in proving confluence for polygraphs presenting 2-categories. For this purpose, we propose a generalization of the usual algorithm for computing critical pairs in term rewriting systems, in which the critical pairs are represented as operations in the operad of contexts in a compact 2-category. [Slides] [Top]

The purpose of this talk is to explain and to generalize, in a homotopical way, the result of Barannikov-Kontsevich and Manin which states that the underlying homology groups of some Batalin-Vilkovisky algebras carry a Frobenius manifold structure. To this extent, we first make the minimal model for the operad encoding BV-algebras explicit. Then we extend the action of the homology of the Deligne-Mumford moduli space of genus 0 curves on the homology of some BV-algebras to an action via higher homotopical operations organized by the cohomology of the open moduli space of genus 0 curves. Applications in Poisson geometry and Lie algebra cohomology and to the Mirror Symmetry conjecture will be given. [Based on the joint paper ArXiv:1105.2008 with Gabriel Drummond-Cole.] [Top]

A totally compatible dialgebra, resp. totally compatible Lie dialgebra, is a vector space with two binary operations that satisfy individual and mixed associativity conditions, resp. Lie algebra conditions. We show that totally compatible dialgebras are closely related to bimodule algebras and semi-homomorphisms. More significantly, Rota-Baxter operators on totally compatible dialgebras provide a uniform framework to generalize known results that Rota-Baxter related operators give tridendriform algebras. Free totally compatible dialgebras are constructed. We also show that a Rota-Baxter operator on a totally compatible Lie dialgebra gives rise to a post-Lie algebra, generalizing the fact that a Rota-Baxter operator on a Lie algebra gives rise to a post-Lie algebra. [Slides] [Top]