This week follows the conferences Operads and Rewriting (Lyon, 2011) and Algebra and Computation (Marseille, 2012), in order to promote the developement of computational methods in algebra and the algebraic formalisation of computation.
Since the eighties, several results (Anick, Squier, Kobayashi, Brown, Dehornoy-Lafont, …) have shown the interest of methods coming from rewriting theory to solve algebraic decision problems or to compute homological invariants. Convergent (confluent and terminating) presentations, and Gröbner bases for linear structures, allow in particular the computation of normal forms and, thus, one can address decision problems (Dehn's problems, combinatorics of Coxeter and Artin groups, …) through computational methods.
The week will cover themes including, but not limited to, the following ones: Garside calculus and the study of Artin groups, diagram algebras in representation theory, rewriting methods in operad theory.
|11h - 12h30||Patrick Dehornoy||Garside calculus I|
|14h30 - 15h30||Luis Paris||Different aspects of the Salvetti complex|
|15h45 - 16h45||Eric Hoffbeck||Leibniz homology as functor homology|
|17h15 - 18h15||Pierre-Louis Curien|| A language for the wirings of operads,
cooperads and dioperads
|9h15 - 10h45||Catharina Stroppel||Diagram calculus I|
|11h15 - 12h45||Patrick Dehornoy||Garside calculus II|
|14h30 - 15h30||Bérénice Oger|| From the cohomology of the motion group of the
trivial link to the homology of the hypertree poset
|15h45 - 16h45||Anne-Laure Thiel||Diagrammatic categorification|
|17h15 - 18h15||Vladimir Verchinine|| Two generalizations of braids: inverse
braid monoid and virtual braids
|9h15 - 10h45||Patrick Dehornoy||Garside calculus III|
|11h15 - 12h45||Catharina Stroppel||Diagram calculus II|
|9h15 - 10h45||Catharina Stroppel||Diagram calculus III|
|11h15 - 12h45||Patrick Dehornoy||Garside calculus IV|
|14h30 - 15h30||Michael Ehrig||Graded Brauer algebras and categorification|
|15h45 - 16h45||Alexis Virelizier||3-dimensional HQFTs|
|17h15 - 18h15||Roland Berger||Confluence: an approach adapted to Koszul algebras|
|9h - 10h30||Catharina Stroppel||Diagram calculus IV|
|11h - 12h||Vladimir Dotsenko||Commutative and noncommutative shuffle algebras|
The aim of the minicourse will be to show that a large part of the algebraic properties of braid groups, including the homological approach of Squier, can be extended to more general structures called Garside families. The common feature that is central in this area is the existence of a certain type of distinguished decompositions (“greedy normal form”) for the elements of the considered structures (which can be monoids, groups, categories, or groupoids). The minicourse will emphasize the geometrical aspects in the computation of normal decompositions and the resulting properties from a viewpoint of geometric and homological group theory.
Confluence: an approach adapted to Koszul algebras
It is well known that any quadratic algebra (i.e. an algebra defined by homogeneous quadratic relations) which is confluent w.r.t. a linearly ordered basis of generators, is Koszul (Priddy's theorem). We present in this talk a universal finite dimensional algebra (called confluence algebra) encoding the confluence of all the quadratic algebras through its representation theory. We show that a remarkable idempotent of the confluence algebra, appropriately represented, provides an explicit homotopy of the Koszul complex, recovering in particular Priddy's theorem. Above the confluence algebra, another finite dimensional algebra (called reduction algebra) allows to take into account the nonconfluent algebras as well, and to show how to repair the confluence. Our approach of confluence can be applied to N-Koszul algebras, but the construction of an homotopy of the Koszul complex is not clear when N>2.
A language for the wirings of operads, cooperads and dioperads
In my work with Hugo Herbelin, and then with my student Guillaume Munch, we explored term syntaxes for sequent calculus (a formalism for writing formal proofs proposed by Gentzen in the 1930s). I recently found out that the same style of syntax allows for a neat description of the wiring structures underlying operads, cooperads, and dioperads. Under this new light, the fundamental non-determinism of classical logic appears to match (a coloured version of) Loday's duplicial algebras.
Commutative and noncommutative shuffle algebras
Shuffle algebras are a mild generalisation of associative algebras: they are left modules (in the category of symmetric collections) over the associative operad (while usual associative algebras are left modules that are concentrated in arity zero). I shall talk about how to use these algebras and their commutative versions to encode some questions of combinatorics and representation theory, and how to work with these algebras in order to answer the respective questions.
Graded Brauer algebras and categorification
The Brauer algebra is a classical diagram algebra that generalizes the symmetric group. It occurs as the centralizer algebra for symplectic and orthogonal Lie algebras as well as certain Lie superalgebras. In this talk I want to discuss how one can use categorification techniques involving generalized Khovanov algebras (discussed in the minicourse by Catharina Stroppel) and the parabolic category O to obtain graded versions of Brauer algebras. This is joint work with Catharina Stroppel.
Leibniz homology as functor homology
For a Lie algebra, Leibniz homology is a non-commutative version of Chevalley-Eilenberg homology. In this talk, we will show how to write this homology theory as functor homology, that is a Tor functor over a category of functors. This extends results of Pirashvili, Richter, Robinson and Whitehouse for associative or commutative algebras. We will insist on the key role of rewriting. This is a joint work with Christine Vespa.
From the cohomology of the motion group of the trivial link to the homology of the hypertree poset
In 2004, J. McCammond and J. Meier have used the hypertree poset to compute the cohomology of the motion group of the trivial link, also called the pure symmetric automorphisms group. They gave a presentation for the cohomology algebra of this group, that we used to try to show the Koszulness of the algebra. However, this Koszulness has recently been disproved by A. Conner and P. Goetz.
J. McCammond and J. Meier also computed the Möbius number of the hypertree poset, which has its homology concentrated in highest degree. As the dimension of the only homology group of the poset of hypertrees on n vertices is the same as the dimension of the vector space of degree n-1 associated with the PreLie operad, we use the theory of species to compute the action of the symmetric group on the first space and compare it to the second space.
Different aspects of the Salvetti complex
The Salvetti complex is originally a cellular complex associated to a finite collection of hyperplanes in a real vector space. Its main property is that it has the same homotopy type as the complement of the complexifications of these hyperplanes. Since its introduction in 1987 by Mario Salvetti, a simplicial version has been defined by Salvetti and, independently, by myself in 1993, and a version for Artin groups has been defined by Charney and Davis in 1995. Recently, I introduced a version that unifies all known versions, valid for infinite collections of hyperplanes as well as for Artin groups. The goal of this talk is to present this new version of the Salvetti complex, placing it in its historical context, and indicating some applications such as in the computation of homologies of Artin groups and of hyperplane arrangements.
In a joint work with Marco Mackaay, we categorify the extended affine type A Hecke algebra and the affine quantum Schur algebra S(n,r) for 2 < r < n, using Elias-Khovanov and Khovanov-Lauda type diagrams. We also define an extension of the 2-categories of affine (singular) Soergel bimodules and construct the affine analogue of the Elias-Khovanov and the Khovanov-Lauda 2-representations of our diagrammatic categorifications into these bimodules 2-categories. In this talk, the aim is to present categorification and in particular its diagrammatic approach, which has the benefit of working with more handable categories but also provides descriptions by generators and relations of abstract categories. We will illustrate this in details by describing both the extended affine Soergel category and its diagrammatic counterpart, which categorify the extended Hecke algebra of affine type A. If time permits, we will treat the case of the affine Schur algebra.
Two generalizations of braids: inverse braid monoid and virtual braids
Inverse braid monoid describes a structure on braids where the number of strings is not fixed. So, some strings of initial n may be deleted. In the talk we show that many properties and objects based on braid groups may be extended to the inverse braid monoids. Namely, there exists an inclusion into a monoid of partial monomorphisms of a free group. This gives a solution of the word problem. Another solution is obtained by an approach similar to that of Garside.
Another generalization of classical braids is the group of virtual braids. We discuss various properties of the pure virtual braid group on three strands. Out of its presentation, we get a free product decomposition for this group. As a consequence, we show that it is residually torsion free nilpotent. Moreover we prove that the presentation of this group is aspherical. We describe also the cohomology ring and the associated graded Lie algebra.
Homotopy quantum field theory (HQFT) is a branch of quantum topology concerned with maps from manifolds to a fixed target space. The aim is to define and to study homotopy invariants of such maps using methods of quantum topology. I will focus on 3-dimensional HQFTs with target the Eilenberg–MacLane space K(G, 1) where G is a discrete group. (The case G = 1 corresponds to more familiar 3-dimensional TQFTs.) These HQFTs provide numerical invariants of principal G-bundles over closed 3-manifolds which can be viewed as ``quantum'' characteristic numbers. To construct such HQFTs, the relevant algebraic ingredients are G-graded categories, which are monoidal categories whose objects have a multiplicative G-grading.