Andrea Solotar: Tamarkin-Tsygan calculus for gentle algebras

24th July, 2023

The whole structure given by the Hochschild cohomology and homology of an associative algebra A together with the cup and cap products, the Gerstenhaber bracket and the Connes differential is called the Tamarkin-Tsygan calculus. It is invariant under derived equivalence and if we can compute all these invariants provides a lot of information. The calculation of the whole Tamarkin-Tsygan calculus is very difficult and generally not even possible for particular algebras. However, there exist some calculations for individual algebras. The problem is, in general, that the minimal projective bimodule resolutions are difficult to find and even if one is able to compute such a resolution, it might be so complicated that the computation of the Tamarkin-Tsygan calculus is not within reach. For monomial algebras the minimal projective bimodule resolution is known and in the case of quadratic monomial algebras it is simple enough, to embark on the extensive calculations of the Tamarkin Tsygan calculus. Yet even for quadratic monomial algebras, the combinatorial level of the calculations is such that it is too complicated to calculate the whole calculus. On the other hand for gentle algebras, the additional constraints on their structure are such that the calculations become possible. We will focus on the concrete aspects of these calculations.

Sigiswald Barbier: Diagram categories of Brauer type

3rd July, 2023

Diagram categories are a special kind of tensor categories that can be represented using diagrams. In this talk I will give an introduction to categories represented using Brauer diagrams. In particular I will explain the relation with the Brauer algebra and how the categorical framework can be applied to representation theory of the corresponding algebra.

Corey Jones: K-theoretic classification of fusion category actions on locally semisimple algebras

14th May, 2023

An action of a tensor category C on an associative algebra A is a linear monoidal functor from C to the monoidal category of A-A bimodules. We consider the problem of classifying (unitary) actions of (unitary) fusion categories on inductive limits of semisimple associative algebras (called locally semisimple algebras). A theorem of Elliot classifies locally semisimple algebras by their ordered K0 groups. We extend this theorem to a K-theoretic classification of fusion category actions on locally semisimple algebras which have an inductive limit decomposition.

Kang Lu: A Drinfeld presentation of twisted Yangians

12th May, 2023

In this talk, I will discuss the Drinfeld’s new presentation for (Olshanski’s) twisted Yangians of type AI that is open for 30 years. This presentation comes from the Gauss decomposition and turns out to be compatible with the one obtained from degeneration of affine iQuantum Groups. If time permits, I will also discuss our work on twisted Yangians of other types.

Toshiki Nakashima: Categorified crystal structure on quantum coordinate rings and cellular crystals

12th May, 2023

For the quiver Hecke algebra R associated with a simple Lie algebra, let R-gmod be the category of finite-dimensional graded R-modules. It is well known that it categorifies the unipotent quantum coordinate ring 𝒜q, that is, the Grothendieck ring 𝒦(R-gmod) is isomorphic to 𝒜q. For the localization of R-gmod, denoted by R̃-gmod, its Grothendieck ring 𝒦(R̃-gmod) defines the localized (unipotent) quantum coordinate ring 𝒜̃q. We shall give a certain crystal structure on the localized quantum coordinate ring 𝒜̃q by regarding the set of self-dual simple objects 𝔹(R̃-gmod) in R̃-gmod.

We also give the isomorphism of crystals from 𝔹(R̃-gmod) to the cellular crystal 𝔹i=Bi1⊗ . . . ⊗BiN for an arbitrary reduced word i=i1 . . . iN of the longest Weyl group element. This result can be seen as a localized version for the categorification of the crystal B(∞) by Lauda-Vazirani since the crystal B(∞) is realized as a subset of the cellular crystal 𝔹i.

Marion Boucrot: The relation between A∞-morphisms and pre-Calabi-Yau morphisms

2nd May, 2023

Pre-Calabi-Yau algebras were introduced in the last decade by  M. Kontsevich, A. Takeda and Y. Vlassopoulos using the necklace bracket. This notion is equivalent to a cyclic A∞-algebra for the natural bilinear form in the finite-dimensional case. Moreover, W-K. Yeung showed that double Poisson DG structures provide an example of pre-Calabi-Yau structures. In 2020, D. Fernandez and E. Herscovich proved that given a morphism of double Poisson DG algebras from A to B, one can produce a cyclic A∞-algebra and A∞-morphisms between the latter and the cyclic A∞-algebras associated to A and B. I will explain how to generalize this result to pre-Calabi-Yau algebras by doing an explicit construction of a (cyclic) A∞-algebra and A∞-morphisms given a pre-Calabi-Yau morphism.

Kent Vashaw: Twisting of universal quantum groups and comodule algebras

1st May, 2023

We consider 2-cocycle twists (and more generally, Morita-Takeuchi equivalences between) Manin's universal quantum groups and their comodule algebras. We show when Zhang twists of connected graded algebras can be realized as cocycle twists, thus concretely connected the (graded) representation theory of an algebra A to the corepresentation theory of its universal quantum group. We also prove that fundamental properties of non-commutative associative algebras, such as Artin-Schelter regularity and Koszuality are preserved under 2-cocycle twist.

Thiago Castilho de Mello: Images of polynomials on algebras

25th April, 2023

The so-called Lvov-Kaplansky Conjecture states that the image of a multilinear polynomial evaluated on the matrix algebra or order n is always a vector subspace. A solution to this problem is known only for n=2. In this talk we will present analogous conjectures for other associative and non-associative algebras and for graded algebras. Also, we will show how we can use gradings to present a statement equivalent to the Lvov-Kaplansky conjecture.

Milen Yakimov: On the spectrum and support theory of a finite tensor category

4th April, 2023

Finite tensor categories are important generalizations of the categories of finite-dimensional modules of finite-dimensional Hopf algebras. There are two support theories for them, the cohomological one and one based on the noncommutative Balmer spectrum of the corresponding stable module category. We will describe general results linking the two types of support via a new notion of categorical center of the cohomology ring of a finite tensor category and will state a conjecture giving the exact relation. The construction and results will be illustrated with various examples.

Hamid Usefi: Polynomial identities, group rings and enveloping algebras

28th March, 2023

I will talk about the development of the theory of polynomial identities initiated by important questions such as Burnside's asking if every finitely generated torsion group is finite. The field was enriched by contributions of many great mathematicians. Most notably Lie rings methods were developed and used by Zelmanov in the 1990s to give a positive solution to the restricted Burnside problem which awarded him the Fields medal. It has been of great interest to expand the theory to other varieties of algebraic structures. In particular, I will review when a group algebra or enveloping algebra satisfy a polynomial identity.