Ottawa-Carleton Joint Algebra Seminar

To every graded Frobenius superalgebra, we associate an algebra that we call an affine wreath product algebra. These algebras appear naturally in categorification. Special cases include degenerate affine Hecke algebras, affine Sergeev algebras (also known as degenerate affine Hecke-Clifford algebras), and wreath Hecke algebras. We investigate the structure of these algebras, describing explicit bases, centers, and Jucys-Murphy elements. We also classify the simple modules and investigate cyclotomic quotients. Specializing the choice of graded Frobenius superalgebra recovers known results (sometimes with more direct proofs) and also proves some open conjectures. In this way, we see that the setting of affine wreath product algebras is a powerful generalization and unification of many important algebraic structures appearing in the literature.

Given a commutative square with the top arrow surjective, the Snake Lemma produces a six term exact sequence, constructed from the kernel and cokernel of the morphisms involved. Its version for pointed groupoids, also called the Brown sequence, constructs a six term exact sequence from a fibration, its kernel and their homotopy invariants. We generalize this result for a *-fibration internal to a regular pointed category. Through the normalization process, we get back the classical Snake Lemma.

Due to the assumption of surjectivity, the Snake Lemma is somehow asymmetric. To overtake this asymmetry, Vitale established the so called Snail Lemma, which associates a six term exact sequence from any commutative square. In the case where the top arrow is a surjection, we get back the Snake Lemma. Its denormalised version, also called the Gabriel-Zisman sequence, constructs from any functor F between internal groupoids an exact sequence using the strong homotopy kernel of F. With a careful analysis of *-fibrations and the comparison between kernels and strong homotopy kernels, one can deduces the Brown sequence from the Gabriel-Zisman sequence.

This is a joint work with Sandra Mantovani, Guiseppe Metere and Enrico Vitale.

The theory of local and global Weyl modules for the standard maximal parabolic subalgebra of an affine Lie algebra is well developed and has many interesting applications. In this talk, we discuss the theory for an arbitrary maximal parabolic subalgebra of an affine Lie algebra. We see that such subalgebras can be thought of as arising in a natural way from a Borel–de Siebenthal pair of semsimple Lie algebras. They are also an interesting set of examples of equivariant map algebras defined by Neher, Savage and Senesi. We shall see that there are examples of non-trivial global Weyl modules which are irreducible and finite-dimensional. Finally for certain good subalgebras we show that the endomorphism ring of a global Weyl module is a Stanley–Reisner ring.

We will first review some history and recent topics. Then, we will discuss Kac-Moody groups over fields as well as Chevalley groups over Dedekind domains. Finally we will deal with simplicity and Galois descent for Kac-Moody groups.

Heegaard Floer theory is a suite of invariants for studying low-dimensional manifolds. In the case of punctured torus, for instance, this theory constructs a particular algebra. And, the invariants associated with three-manifolds having (marked) torus boundary are differential modules over this algebra. This is structurally very satisfying, as it translates topological objects into concrete algebraic ones. I will discuss a geometric interpretation of this class of modules in terms of immersed curves in the punctured torus. This point of view has some surprising consequences for closed three-manifolds that follow from simple combinatorics of curves. For example, one can show that, if the dimension of an appropriate version of the Heegaard Floer homology (of a closed manifold) is less than 5, then the manifold does not contain an essential torus. Said another way, this gives a certificate that the manifold admits a geometric structure à la Thurston. This is joint work with Jonathan Hanselman and Jake Rasmussen.

Extended affine Lie algebras are a class of Lie algebras that includes simple complex Lie algebras, affine Kac-Moody algebras, toroidal Lie algebras and many more types of Lie algebras. As in these examples, the existence of Cartan subalgebras is a crucial part of the structure of an extended affine Lie algebra, and as in the examples conjugacy of Cartan subalgebras in extended affine Lie algebras is a fundamental question. In this talk I will discuss the conjugacy problem. The talk is based on joint work with Chernousov and Pianzola.

Very little has been known about representation theory of Lie algebras of polynomial vector fields on affine algebraic varieties beyond the cases of affine space and a torus. We study a category of representations of the Lie algebras of vector fields on algebraic variety X that admit a compatible action of the algebra of polynomial functions on X. We construct simple modules in this category and state a conjecture on the general structure of such modules. This is a joint work with Slava Futorny and Jonathan Nilsson.

Goresky, Kottwitz and MacPherson showed that the equivariant cohomology of varieties equipped with an action of a torus T can be described using the so called moment graph, hence, translating computations in equivariant cohomology into a combinatorial problem. Braden and MacPherson proved that the information contained in this moment graph is sufficient to compute the equivariant intersection cohomology of the variety. In order to do this, they introduced the notion of a sheaf on moment graph whose space of sections (stalks) describes the (local) intersection cohomology. These results motivated a series of paper by Fiebig, where he developed and axiomatized sheaves of moment graphs theory and exploited Braden-MacPherson’s construction to attack representation theoretical problems.

In the talk we explain how to extend the theory of sheaves on moment graphs to an arbitrary algebraic oriented equivariant cohomology h in the sense of Levine-Morel (e.g. to K-theory or algebraic cobordism). Moreover, we show that in the case of a total flag variety X the space of global sections of the respective h-sheaf also describes an endomorphism ring of the equivariant h-motive of X.

This is a joint work with Rostislav Devyatov and Martina Lanini.

Let Φ=Φ(X) be a free abstract group on the finite set X = {x₁,...x_r}. Let g be an element of Φ and let S be a subset of Φ. The main topic of this talk is to describe algorithms to decide whether or not g is an element of S, when S is 'reasonable' and it is described in a reasonable manner. We consider the cases when S is either a finitely generated subgroup H of Φ or a product H₁,...,H_m, where each H_i is a finitely generated subgroup of Φ or topological closures of such type of subsets with respect to certain natural topologies on Φ.

In this talk, we will briefly review the basic theory of Demazure modules, which are modules over the standard maximal parabolic subalgebra of an affine Lie algebra. The discussion will be followed by some connections that I have discovered (with collaborators) in my own research between algebraic combinatorics and other areas of mathematics such as representation theory and number theory. For instance, we show that the graded multiplicities of higher level Demazure modules in Demazure flags can be expressed in terms of Dyck paths. The generating series for those graded multiplicities give rise to interesting connections with Ramanujan's mock theta functions. I will describe some results and further questions in this direction.