Vertex operator algebra (VOA) V with a Virasoro element L is called modular invariant if its normalized character ch_V (tau):=tr_V e^{2pi i tau (L_0 -c/24)} is a modular invariant function. The basic examples are rational VOA. In my talk I will present a number of examples of modular invariant VOA beyond the rational ones, and state several conjectures and open problems about them. In conclusion I will discuss the quasi-invariant VOA, for which the characters are quasi-modular functions. The examples include the simply-laced affine VOA  at negative integer level \geq -b, where b is the length of the longest leg in the affine Dynkin diagram.

n--Cayley--Hamilton algebras were introduced in 1987 by Procesi  to characterize  associative algebras which embed into n x n matrices  over commutative rings.

They are defined as associative algebras with a trace which satisfy the n—characteristic polynomial formally deduced from the trace.

At that time the theory was only in characteristic 0 due to several technical obstacles.  It was proved that an n--Cayley--Hamilton algebra has a natural embedding  into n x n matrices  over a universal commutative ring.

Later the theory was also formalized in a characteristic free form, replacing the axioms of a trace with axioms for a norm,  the question of  the embedding remains open.  I will discuss some experimental work done using group algebras of 2-groups in characteristic 2.

In this seminar, I will discuss some problems encountered jointly with Luca Casarin in our study of a factorizable version of the Feigin-Frenkel theorem regarding the center of the enveloping algebra. In this work, algebraic objects are often equipped with a topology that is important to consider; it would be desirable for natural algebraic constructions to preserve certain good topological properties, such as completeness. Given the level of generality of our work, the interplay between algebraic and topological properties forced us to address a series of additional issues. The seminar will focus on these somewhat technical aspects of the work done with Luca Casarin, highlighting some small differences required in the definition of the fundamental objects.

(joint work with G. A. García and M. Paolini)

Several version of "multiparameter quantum groups" (mostly in the form of QUEA's) have been introduced in literature.  A standard recipe for that amounts to starting from a uniparameter QUEA and then modifying it via a process of "deformation" (in a Hopf-theoretical sense): in turn, the latter exists in two versions, by twist and by 2-cocycle, that affect either the multiplicative or the comultiplicative structure alone.  Indeed, most examples known so far in literature happen to fall within such a framework.

I will present a far-reaching extension of this idea, which apply to all QUEA's whose underlying Lie algebra is a symmetrisable Kac-Moody.  Surprisingly enough, it turns out that the two families of multiparameter QUEA obtained via deformation (by twist or by 2-cocycle) actually coincide; this whole, unique family then is stable under deformations of either type.

I will also present the parallel construction leading, independently, to multiparameter Lie bialgebras (in short, MpLbA’s): this provides a broad family of MpLbA's, that again is stable for deformations (either by twist or by 2–cocycle). The semiclassical limit of any multiparameter QUEA as above is a MpLbA, and conversely each MpLbA can be quantized to some multiparameter QUEA. Moreover, the processes of “specialization” and of “deformation” commute with each other.

Time permitting, I will shortly touch upon the generalisation of all the above to the setup of Lie superbialgebras and of quantum supergroups.

All this comes from joint work with G. A. García and M. Paolini.

Let g be a semisimple Lie algebra over the complex numbers, with a fixed Borel subalgebra with nilradical n. Given a nilpotent orbit in g, its intersection with n decomposes into several irreducible components, all having the same dimension. An orbital variety in n is by definition such an irreducible component, for some nilpotent orbit in g. Following Steinberg and Joseph, there is a surjective map from the Weyl group W of g to the set of the orbital varieties: the fibers of this map provide geometric analogues of the Kazhdan-Lusztig cells in W. Orbital varieties are naturally ordered by the inclusion of their closures. In the talk I will consider orbital varieties coming from the spherical nilpotent orbits, and I will discuss some questions and conjectures relating them to suitable fully commutative elements in the corresponding cells.

An involution system (W,S) is a pair where W is generated by a set of involutions S. Any involution system is naturally endowed with a weak order arising from orienting its Cayley graph. In the case (W,S) is a Coxeter system, Björner showed that the weak order is a complete meet-semilattice. This fact has many important consequences for Coxeter systems and their connected structures. In this article, we discuss the following question: For which involution systems the weak order is a complete meet-semilattice?

I will first discuss examples of involution systems that satisfies the condition to be a complete meet-semilattice (and that are not Coxeter systems). Then, in the case of an involution system with sign character, I will discuss their presentation by generators and relations and a classification in rank 3. If time allows, I will also discuss open problems in relation to automatic structures and geometric representations.

(Joint work with Fabricio Dos Santos and Aleksandr Trufanov)

We will describe all Verma modules for the infinite-dimensional linearly compact Lie superalgebra E(4,4) which are not irreducible, and we will show that all such modules can be arranged in a beautiful picture, consisting of infinitely many sequences of modules.

In this talk we discuss some applications of the theory of conformal embeddings and collapsing levels for affine vertex algebras developed in joint papers with Kac, Moseneder, Papi,  Perse. We present a proof of the semi-simplicity of the Kazhdan-Lusztig category KL of affine vertex superalgebras at collapsing and some other levels. The proof uses the representation theory of affine vertex algebras and concepts from the theory of conformal embeddings. In the Lie superalgebra case, we discuss some examples when KL_k has indecomposable highest weight modules and explains what is a possible implication of this in the representation theory of vertex algebras. We will also discuss relations of our results to vertex algebras with origins in some physics theories.