Schedule for: 16w5070 - Vertex Algebras and Quantum Groups

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk I will survey some of the basic properties of the category of finite dimensional tilting modules for the quantized enveloping algebra of a simple Lie algebra. At a generic parameter this category is semisimple and consists of all finite dimensional modules. But at roots of unity the category is non-semisimple and has a rich structure. We shall highlight some of its main properties and also point to several applications. At the end we shall discuss some recent developments and a couple of open questions.

Quantum groups are a fertile area for explicit computations of cohomology and support varieties because of the availability of geometric methods involving complex algebraic geometry. Ginzburg and Kumar have shown that for l>h (l is order of the root of unity and h is the Coxeter number), the cohomology ring identifies with the coordinate algebra of the nilpotent cone of the underlying Lie algebra g=Lie(G). Bendel, Pillen, Parshall and the speaker have determined the cohomology ring when l is less than or equal to h and have shown that in most cases this identifies with the coordinate algebra of a G-invariant irreducible subvariety of the nilpotent cone. The latter computation employs vanishing results on partial flag variety G/P via the Grauert-Riemenschneider theorem. Support varieties have been determined for tilting modules (by Bezrukavinov), induced/Weyl modules (by Ostrik and Bendel-Nakano-Pillen-Parshall), and simple modules (by Drupieski-Nakano-Parshall). The calculations for tilting modules and simple modules employed the deep fact that the Lusztig Character Formula holds for quantum groups when l>h. In this talk, I will survey several of the main results of the topic and indicate the combinatorial and geometric techniques necessary to make such calculations. Open problems will also be presented.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. Please don't be late, or you will not be in the official group photo! The photograph will be taken outdoors so a jacket might be required.

Bethe ansatz is a physics motivated method which is used to diagonalize matrices which appear as Hamiltonians of various integrable systems. In particular, it can be applied to the case where the matrices have size 1x1. Interestingly, it leads to a variety of non-trivial questions with important applications. In this talk I will review the basics of the Bethe ansatz on the example of the Gaudin model and discuss the results and conjectures related to the 1x1 case.

We shall discuss how a desire to work over singular varieties leads to infinity versions of Picard-Lie algebroids and their vertex/chiral algebra analogues.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Let $V$ be a simple vertex operator algebra and $G$ a finite automorphism group of $V$ such that $V^G$ is regular. It is proved that every irreducible $V^G$-module occurs in an irreducible $g$-twisted $V$-module for some $g\in G.$ Moreover, the quantum dimensions of each irreducible $V^G$-module is determined and a global dimension formula for $V$ in terms of twisted modules is obtained.

Rational VOAs are by now quite well understood: their representation theory is captured by a modular tensor category; their characters define a vector-valued modular form for $SL(2,Z)$; etc. The class of C2-cofinite (logarithmic) VOAs is the natural generalisation of rationality, but their theory is still much less clear. This talk reviews and contributes to this theory. It is joint work with Thomas Creutzig.

We shall first recall explicit realizations of certain affine and superconformal vertex algebras from [D. Adamovic, Transform. Groups (2015)] and study their relations with vertex operator algebras appearing in LCFT. Then we shall consider a generalization motivated by the construction of conformal embeddings of affine vertex algebras in W-algebras. We shall also present a decomposition of a large family of non-rational affine W-algebras as modules for affine vertex operator algebras at admissible and negative levels. A particular emphasis will be put on the application of affine fusion rules and intertwining operators in the determination of branching rules. The second part of this talk is based on a joint paper with V. Kac, P. Moseneder-Frajria, P. Papi and O. Perse.

Speakers: K. Nagatomo, S. Kanade, X. He, A. Zeitlin, J. van Ekeren (short talk) and T. Creutzig (short talk)

The Yangian Yg and quantum loop algebra Uq(Lg) of a complex semisimple Lie algebra g share very many similarities, and were long thought to have the same representations, though no precise relation between them existed until recently. I will explain how to construct a faithful functor from the finite-dimensional representations of Yg to those of Uq(Lg). The functor is entirely explicit, and governed by the monodromy of the abelian difference equations determined by the commuting fields of the Yangian. It yields a meromorphic, braided Kazhdan-Lusztig equivalence between finite-dimensional representations of the Yg and of U_q(Lg). A similar construction yields a faithful functor from representations of U_q(Lg) to those of the elliptic quantum group E_{q,t}(g) corresponding to g. This allows in particular a classification of irreducible finite-dimensional representations of E_{q,tau}(g), which was previously unknown. This is joint work with Sachin Gautam (Perimeter Institute).

We use the Jacobi-Trudi identity to incorporate several well-known families of symmetric functions to uniformly treat generalized Schur symmetric functions and their vertex operator realization. Under the general set-up, we prove that Gambelli identity also holds, thus derive several scattered results under one umbrella. In particular, this includes Weyl's character formulas of classical simple Lie algebras and the shifted Schur symmetric functions studied by Olshanski-Okounkov. This is joint work with Natasha Rozhkovskaya.

In this talk, I will discuss a natural connection of a certain $q$-Virasoro algebra with affine Lie algebras and vertex algebras. To any abelian group $S$ with a linear character, we associate an infinite-dimensional Lie algebra $D_{S}$. When $S=\Z$ with $\chi$ defined by $\chi(n)=q^{n}$ with $q$ a nonzero complex number, $D_{S}$ reduces to the $q$-Virasoro algebra $D_{q}$ which was introduced in \cite{BC}. We also introduce a Lie algebra $\g_{S}$ with $S$ as an automorphism group and we prove that $D_{S}$ is isomorphic to the $S$-covariant algebra of the affine Lie algebra $\widehat{\g_{S}}$. Then we relate restricted $D_{S}$-modules of level $\ell\in \C$ with equivariant quasi modules for the vertex algebra $V_{\widehat{\g_{S}}}(\ell,0)$. Furthermore, we show that if $S$ is a finite abelian group of order $2l+1$, $D_{S}$ is isomorphic to the affine Kac-Moody algebra of type $B^{(1)}_{l}$. This talk is based on a joint work with Hongyan Guo, Shaobin Tan and Qing Wang.

Chiral algebras are extensively studied in many areas of both mathematics and physics, due to the wealth of examples from various classes of algebras generated by chiral fields (although precise axiomatic/mathematical definition of the concept is lacking in it full generality). Super vertex algebras constitute a class of chiral algebras, corresponding to the chiral part of a conformal quantum field theory on the complex plane, and their theory is well established. Nevertheless there is a variety of important examples of algebras generated by chiral fields that cannot be described by the concept of super vertex algebra. The most challenging case concerns the quantum vertex operators and the quantum chiral algebras they generate. This area of research, which by now is quite large and growing, was started with the fundamental problem posed by Igor Frenkel: to formulate and develop a theory of quantum vertex algebras incorporating as examples the Frenkel-Jing quantum vertex operators realizing the quantum affine algebras. This problem is still ultimately unsolved despite the comparative progress lately. In this talk we will discuss some of the issues that are encountered on the way to defining a suitable theory of quantum chiral algebras. As a guiding principle for the the mathematical description of any class of chiral algebras we will discuss the notable instances of certain special isomorphisms between chiral algebras of that class (such as the boson-fermion correspondences).

The orbifold and coset constructions are standard ways to create new vertex algebras from old ones. It is believed that orbifolds and cosets will inherit nice properties such as strong finite generation, C_2-cofiniteness, and rationality, but few general results of this kind are known. I will discuss how these problems can be studied systematically using ideas from classical invariant theory. This is based on joint work with T. Creutzig.

I will introduce new twisted Yangians associated to symmetric pairs of types B, C and D which are similar to the twisted Yangians of type A introduced by G. Olshanski around twenty-give years ago and which have been quite well studied. After a discussion of a number of their properties, I will present classification results for their irreducible finite dimensional modules. This is joint work with Vidas Regelskis and Curtis Wendlandt.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.