Titles and abstracts

• Gwyn Bellamy (University of Glasgow): Filtered Koszul duality 

Abstract: In this talk I will revisit Koszul duality for filtered algebras U whose associated graded algebra A is Koszul. In this setting, duality is an equivalence between the derived category D(U) of U and the homotopy category of injective curved dg-modules for the Koszul dual algebra A^!. Here the curvature element corresponds to the deformation data for U. Examples include the Weyl algebra, symplectic reflection algebras and deformed preprojective algebras. If there is time I will explain applications of this result, such as an explicit description of the "exotic" t-structure on the homotopy category of injective curved dg-modules one gets via this equivalence. This is based on joint work in progress with Simone Castellan and Isambard Goodbody. 

• Kieran Calvert (Lancaster University): Clifford algebra invariants of symmetric pairs and Schur polynomials 

Abstract: Let (g,k) be a symmetric pair of reductive Lie algebras, then g is a direct sum of k and p. Using the Killing form, one can construct the Clifford algebra associated to p. In this talk I will discuss the k invariant subalgebra of this Clifford algebra. The related associated graded algebra is a model for the de Rham cohomology of the symmetric space G/K. A description of the Clifford algebra gives a deformed analogue of the cohomology which reveals a conceptually cleaner method to calculate the cohomology of G/K. In particular, for G/K diffeomorphic to a Grassmannian, I will demonstrate a novel way to calculate the product of Schur polynomials. This is joint work with Karmen Grizelj, Andrey Krutov, Kyo Nishiyama and Pavle Pandžić.  

• Iva Halacheva (Northeastern University): Bethe subalgebras of the Yangian Y(gl(n)) and crystals for tame representations 

Abstract: The Bethe subalgebras of the Yangian Y(gl(n)) form a family of maximal commutative subalgebras parametrized by the Deligne-Mumford compactification of the moduli space M(0,n+2). For a fixed tame representation V of Y(gl(n)), and any point C in the real locus of the parameter space, the corresponding Bethe subalgebra B(C) acts on V with simple spectrum. This results in an unramified covering, whose fiber over C is the set of eigenlines for the action of B(C). I will discuss how each such fiber can be identified with a collection of Gelfand-Tsetlin keystone patterns, which carry a gl(n)-crystal structure, and the monodromy action realized by a type of cactus group. This is joint work with Anfisa Gurenkova and Leonid Rybnikov. 

• Simon Lentner (Universität Hamburg): Categories and algebras appearing in logarithmic conformal field theory 

Abstract: The logarithmic Kazhdan Lusztig correspondence is a conjectural equivalence between the representation theory of quantum groups to the representation theory of certain vertex algebras (an algebraic structure appearing in conformal field theory, which I will gently motivate). I will explain the categorical language and recent results with T. Creutzig and M. Rupert, that have allowed us to settle this conjecture in several rank 1 cases. At the same time the approach produces a rather general type of (non semisimple) modular tensor category, extending a given one, that is a good candidate for the representation theory of a kernel-of-screening vertex subalgebra. This has applications much beyond the original intention of the conjecture. 

• Vanessa Miemietz (University of East Anglia): Induction and restriction of 2-representations 

Abstract: For a finite-dimensional algebra A with a subalgebra B, there is the well-known adjunction between induction and restriction (Ind^A_B , Res^A_B) where Ind^A_B = A\otimes_B -. For 2-representations of 2-categories, an analogue of restriction is straightforward, but a lack of relative tensor product means we need to define induction differently in order to obtain an analogous 2-adjunction between induction and restriction (for finitary 2-representations of fiat 2-categories). I will explain the basics of 2-representation theory and the necessary ingredients to state this 2-adjunction.

• Tomasz Przeździecki (University of Edinburgh): Drinfeld rational functions for quantum affine symmetric pairs 

Abstract: It is well known that quantum affine algebras admit three distinct presentations (Kac-Moody, new Drinfeld and RTT). Relatively recently, the same has been shown to hold for a broad family of quantum affine symmetric pairs. In particular, a Drinfeld-type presentation, due to Lu-Wang, is a new and exciting development.

The focus of my talk will be the relationship between the usual Drinfeld presentation of quantum affine algebras and the Lu-Wang presentation of their coideal subalgebras. Remarkably, both presentations exhibit large commutative subalgebras, which are of particular interest to representation theory. More specifically, I will present several results concerning the properties of the generators of these commutative subalgebras, including their behaviour under inclusion and the coproduct, as well as their spectra on finite-dimensional representations. I will also discuss applications to the nascent theory of q-characters of quantum symmetric pairs. 

• Catharina Stroppel (Universität Bonn): Semistrict monoidal 2-categories of chain complexes, braidings and dg-centralizers 

In this talk we will try to give an idea what a braiding on a  monoidal 2-category is with the main example given by Soergel bimodules. We will explain a  general construction for semistrict monoidal 2-categories with examples. Finally we talk about dg-centers and dg-centralizers and connect this with Drinfeld centres and our desired braidings.