Abstracts

- Drazen Adamovic: On indecomposable and logarithmic modules for affine vertex operator algebras

In the first part of the talk we review our recent explicit realizations of certain affine vertex algebras and discuss their applications in the representation theory. Admissible affine vertex operator algebras $V_{k} (\mathfrak g)$ are semi-simple in the category $\mathcal O$. In this talk, we shall consider $V_{k} (\mathfrak g)$--modules outside of the category $\mathcal O$.

Logarithmic modules appear in the non-split extension of certain weight modules. Although $V_{k} (\mathfrak g)$--modules are modules for the affine Lie algebras, it is difficult to construct indecomposable and logarithmic modules using concepts from the representation theory of Lie algebras. We will show how these modules can be explicitly constructed using vertex-algebraic techniques. We will also show that certain Whittaker modules are also weak $V_{k} (\mathfrak g)$--modules.

- Hadewijch De Clercq: The Askey-Wilson and $q$-Bannai-Ito algebra extended to higher rank

The Askey-Wilson algebra is a quadratic quantum algebra with remarkable properties. It is closely connected to integrable systems and it encodes the bispectrality of the Askey-Wilson polynomials. A similar algebraic underpinning can be given to the Bannai-Ito polynomials. The resulting Bannai-Ito algebra has a $q$-deformation, which was recently constructed in the threefold tensor product of $\mathfrak{osp}_q(1\vert 2)$.

In this talk I will describe and compare the Askey-Wilson and $q$-Bannai-Ito algebra. The Hopf algebra formalism allows to extend both algebras to arbitrary rank, via a novel general algorithm. I will construct a superintegrable model possessing the symmetry of these newly constructed higher rank algebras. The central operator in this model is built from reflections and $q$-analogs of $\mathbb{Z}_2$-Dunkl operators. We will study it through modules of its polynomial null-solutions.

This is joint work with Hendrik De Bie and Wouter van de Vijver.

- Azat Gainutdinov: Quasi-Hopf algebras for extended W-algebras in Logarithmic CFTs

I will talk about representation categories of Vertex Operator Algebras describing certain class of Logarithmic CFTs, in the case when they are rigid and have finitely many (iso classes) of irreducible representations. Such categories are non-semisimple and are expected to be modular tensor categories (i.e. with non-degenerate braiding). Basic examples are provided by the so-called triplet W-algebras and by chiral algebras of N pairs of symplectic fermion fields. Calculating Perron-Frobenius dimensions of irreducible representations, one can conclude that the representation categories in these cases should be realized via representations of a factorizable (quasi-)Hopf algebra. It is indeed the case and turns out that the quasi-Hopf algebras we encounter here are simple modifications of well-know finite-dimensional quantum groups at roots of unity. In particular for the triplet W-algebra, we have proposed a braided monoidal equivalence with the representation category of a quasi-Hopf modification of the restricted quantum group for $sl(2)$ at roots of unity of even order. The main ingredient in the construction of such a quasi-quantum group is the theory of VOA extensions by simple currents. This is a joint work with T. Creutzig and I. Runkel. Using another approach based on theory of Nichols algebras in braided categories, we also found a unique (up to a twist) factorizable quasi-Hopf algebra for any .finite root system and any root of unity of even order, and formulated a conjecture about ribbon equivalence with representation categories of extended W-algebras which are defined as the kernel of screenings in a lattice VOA. This is a joint work with S. Lentner and T. Ohrmann.