Tag - Representation theory

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.

Tiago Macedo: Finite-dimensional modules for map superalgebras

10th July, 2023

In this talk we will present recent results on the category of finite-dimensional modules for map superalgebras. Firstly, we will show a new description of certain irreducible modules. Secondly, we will use this new description to extract homological properties of the category of finite-dimensional modules for map superalgebras, most importantly, its block decomposition.

Jan Trlifaj: Deconstructible classes, approximations, and AECs of modules

4th July, 2023

Deconstructible classes of modules are among the main sources of approximations in relative homological algebra. They also occur in connection with abstract elementary classes (AECs). The latter were introduced by Shelah as far-reaching generalizations of classic first-order structures. A direct connection is provided by the 'AECs of roots of Ext': these are the AECs of the form P = (𝒜,≼) where 𝒜 = { M in Mod-R such that ExtiR(M,N) = 0 for all i > 0 and all N in 𝒞 } for a class of modules 𝒞, and ≼ is a partial order on 𝒜 satisfying X ≼ Y iff Y/X is in 𝒜. P is an AEC iff 𝒜 is a deconstructible class closed under arbitrary direct limits. A major open problem concerning AECs is Shelah's Categoricity Conjecture (SCC). It claims that categoricity of an AEC is a large enough cardinal λ (= existence of a unique structure in 𝒜 of cardinality λ up to isomorphism) is equivalent to its categoricity in a tail of cardinals. After recalling the role of deconstructible classes of modules, we will prove SCC for the AECs of roots of Ext, and more in general, for all 'deconstructible' AECs (𝒟,≤), i.e., such that 𝒟 is a deconstructible class of modules. We will also consider the open problem of whether for all deconstructible AECs, the class 𝒟 is necessarily closed under direct limits. We will show that it is consistent with ZFC that 𝒟 is closed under countable direct limits provided that 𝒟 is closed under direct summands and ≤ refines direct summands.

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.

Lior Bary-Soroker: Random additive polynomials

1st June, 2023

Random polynomials with integer coefficients tend to be irreducible and to have a large Galois group with high probability. This was shown more than a century ago in the large box model, where we choose the coefficients uniformly from a box and let its size go to infinity, while only recently there are results in the restricted box model, when the size of the box is bounded and its dimension (i.e., the degree of the polynomial) goes to infinity. In this talk, we will discuss an important class of random polynomials: additive polynomials, which have coefficients in the polynomial ring over a finite field. In this case, the roots form a vector space, hence the Galois group is naturally a subgroup of GLn. While we prove that the Galois group is the full matrix group both in the large box model, and in the large finite field limit, our main result is in the restricted box model: under some necessary condition the Galois group is large (in the sense that it contains SLn) asymptotically almost surely, as the degree goes to infinity. The proof relies crucially on deep results on subgroups of GLn by Fulman and Guralnick, combined with tools from algebra and number theory.

Kaiming Zhao: Simple smooth modules

29th May, 2023

Let L be a graded Lie algebra by integers with k-th homogenous space Lk where k are integers. An L-module V is called a smooth module if any vector in V can be annihilated by Lk for all sufficiently large k. Smooth modules for affine Kac-Moody algebras were introduced and studied by Kazhdan and Lusztig in 1993. I will show why this class of modules should be studied and what results are known now. An easy characterization for simple smooth modules for some Lie algebras will be provided.

Amnon Yekutieli: A DG Approach to the Cotangent Complex

23rd May, 2023

Let B/A be a pair of commutative rings. We propose a DG (differential graded) approach to the cotangent complex LB/A. Using a commutative semi-free DG ring resolution of B relative to A, we construct a complex of B-modules LCotB/A. This construction works more generally for a pair B/A of commutative DG rings. In the talk, we will explain all these concepts. Then we will discuss the important properties of the DG B-module LCotB/A. If time permits, we'll outline some of the proofs. It is conjectured that for a pair of rings B/A, our LCotB/A coincides with the usual cotangent complex LB/A, which is constructed by simplicial methods. We shall also relate LCotB/A to modern homotopical versions of the cotangent complex.

Malihe Yousofzadeh: Finite Weight Modules over Affine Lie Superalgebras

22nd May, 2023

Nonzero real vectors of an affine Lie superalgebra act on a simple module either locally nilpotently or injectively. This helps us to divide simple finite weight modules over a twisted affine Lie superalgebra L into two subclasses called hybrid and tight. We will talk about the characterization as well as the classification problem of modules in each subclass. In this regard, the classification of bases of the root system of L is crucial. We will discuss how we can classify the bases and how we can use the obtained classification to study simple finite weight modules over L.

Lewis Topley: Modular representation theory and finite W-algebras

15th May, 2023

Finite W-algebras were introduced by Premet in full generality, and they quickly became quite famous for their many applications in the representation theory of complex semisimple Lie algebras, especially the classification of primitive ideals. However, these algebras first appeared in the representation theory of Lie algebras associated to reductive groups in positive characteristic. In this talk I will survey the history of finite W-algebras in modular representation theory, and explain some of the contributions I have made to the field. The main applications in this talk will be the construction and classification of 'small' modules of Lie algebras.