Program

Motivated by the representation theory of affine vertex operator algebras, we study a family of sl₃-modules that are weight with infinite multiplicities. They are proven to be generically irreducible and not Gelfand–Tsetlin with respect to any choice of subalgebra. We also study the composition factors when the modules are reducible.

To appear.

With motivation from noncommutative geometry, we want to know when there exists a *-algebra structure on path algebras of quivers. We consider quivers with anti-involutions, and show that a path algebra admits a *-algebra structure if and only if there exists an anti-involution on its underlying quiver.

In this talk we will discuss modules over the Lie algebra sl(2) that are free of finite rank when restricted to the universal enveloping algebra U(h) of a Cartan subalgebra h of sl(2). In particular, we will present a new family of simple U(h)-free modules of rank 2. The talk is based on a joint work with K. Nguyen and K. Zhao.

The Bernstein-Sato polynomial (or b-function) is a classical invariant in the theory of Weyl algebras that encodes subtle information about singularities of hypersurfaces. In this talk, we introduce a natural generalization of the b-function to modules over associative algebras and use it to study the twisted localization of modules with respect to the Ore set generated by a locally ad-nilpotent regular element. We construct the b-function for a given module and locally ad-nilpotent regular element, and prove that the zeros of this b-function determine the values of the twisting parameter at which the twisted localization of the module is not simple. We illustrate the concept of the b-function for modules over Weyl algebras, universal enveloping algebras of semisimple and affine Lie algebras, and rational Cherednik algebras.

To appear.

I will discuss compatible Lie algebras. These algebras arose from the related class of compatible Poisson algebras in the context of mathematical physics and Hamiltonian mechanics and are related to integrable systems, the classical Yang—Baxter equation and homogeneous subalgebras of loop algebras.

In this talk, we start by stating some basic definitions and results about compatible Lie algebras. We present counterexamples to analogues of the theorems of Weyl and Levi for Lie algebras. Moving to the representation theory of a family of simple two-dimensional compatible Lie algebras, we construct an infinite family of irreducible representations and prove a Clebsch-Gordan type formula. We finish by discussing the failure of further central results from Lie algebra theory, including the characterization of semisimple algebras and the Whitehead Lemmas.

This is joint work with Xabier García Martínez, Manuel Ladra and Bernardo Leite da Cunha.

Several small noncommutative algebras can be realized as generalized Weyls algebras (GWAs). In this talk I will discuss their representation theory, focusing on a new family of modules that are free over the base ring. The talk is based on joint work with Samuel Lopes.

We generalize the Huerfano and Khovanov notion of zigzag algebras in order to determine a representation type of special representations of Jordan algebras with radical square zero. This is joint with V. Bekkert and V. Serganova.

In this talk, I will discuss Zhu algebras associated with permutation orbifold vertex operator algebras, along with some related topics.

I will report some recent works joint with Haowu Wang and Brandon Williams on the hyperbolization of affine Lie algebras and affine Lie superalgebras. This is to classify the Borcherds-Kac-Moody algebras whose denominator is a reflective Borcherds product of singular weight. We find the classification is closely related to some special vertex operator (super)algebras including c=24 holomorphic VOA, c=12 holomorphic VOSA and the conjectural Z-graded c=24 holomorphic VOSA.

In their seminal work on the representation theory of enveloping algebras of finite-dimensional Lie algebras, Dixmier and Moeglin showed that primitive ideals in these algebras could be characterised both algebraically and topologically among the prime ideals. Since then, algebras for which these characterisations of primitive ideals hold are referred to as having the Dixmier-Moeglin Equivalence. Goodearl extended this notion to the Poisson setting and the first examples of Poisson algebras for which the Poisson Dixmier-Moeglin Equivalence (PDME for short) does not hold were obtained in 2014. In this talk, after reviewing the relevant concepts, I will discuss a strong version of the PDME. In particular, I will present several classes of Poisson algebras for which the strong PDME holds, and will explain how we can use orthogonal polynomials to construct Poisson algebras for which the PDME holds but the strong one fails.

This is a report on the PhD work of Nirina Albert Razafimandimby (Porto) under the co-supervision of Sam Lopes (Porto) and myself.

The theory of algebra objects and their module objects internal to monoidal category C is crucial towards understanding the structure of C-module categories. This theory can be viewed as an analogue of the Morita theory of finite-dimensional algebras, but internal to C. In joint work with Tony Zorman (University of Hamburg) and Kevin Coulembier (University of Sydney), we introduced an analogue of the Jacobson radical for an algebra object. Building on this idea, I will present tensor- and fusion-categorical generalizations of classical results of Gabriel and Dlab-Ringel, showing that any algebra object is Morita equivalent to a quotient of a path algebra of a species internal to C. This is work in progress, joint with Edmund Heng (University of Sydney).