Tag - Algebra

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.

Alexey Petukhov: Witt Lie algebra and the associated primitive ideals

17th July, 2023

In my talk I would like to discuss my joint articles with S. Sierra about the primitive ideals of universal enveloping U(W) and the symmetric algebra S(W) of Witt Lie algebra W and similar Lie algebras (including Virasoro Lie algebra). The key theorem in this setting is that every nontrivial quotient by a two-sided ideal of U(W) or S(W) has finite Gelfand-Kirillov dimension. Together with Sierra we enhanced this statement to the description of primitive Poisson ideals of S(W) in terms of certain points on the complex plane plus a few parameters attached to these points. In the end I will try to explain how all these concepts works for the ideals whose quotient has Gelfand-Kirillov dimension 2.

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.

Esther García González: Nilpotent last-regular elements

26th June, 2023

We say that an element x in a ring R is nilpotent last-regular if it is nilpotent of certain index n+1 and its last nonzero power xn is regular von Neumann, i.e., there exists another element y∈R such that xnyxn=xn. This type of elements naturally arise when studying certain inner derivations in the Lie algebra Skew(R,∗) of a ring R with involution ∗ whose indices of nilpotence differ when considering them acting as derivations on Skew(R,∗) and on the whole R. When moving to the symmetric Martindale ring of quotients Qms(R) of R we still obtain inner derivations with the same indices of nilpotence on Qms(R) and on the skew-symmetric elements Skew(Qms(R),∗) of Qms(R), but with the extra condition of being generated by a nilpotent last-regular element. This condition strongly determines the structure of Qms(R) and of Skew(Qms(R),∗). We will review the Jordan canonical form of nilpotent last-regular elements and show how to get gradings in associative algebras (with and without involution) when they have such elements.

Nathan Brownlowe: Self-similar quantum groups

23rd June, 2023

In this talk I will introduce the notion of self-similarity for compact quantum groups. I will start by looking at the quantum automorphism group of an infinite homogeneous rooted tree. Self-similar quantum groups are then certain quantum subgroups of these quantum automorphisms. I will then look at a class of examples called finitely-constrained self-similar quantum groups, and I will describe a subclass as quantum wreath products by subgroups of the quantum permutation group.

Alina Vdovina: Higher structures in Algebra, Geometry and C*-algebras

23rd June, 2023

We present buildings as universal covers of certain infinite families of CW-complexes of arbitrary dimension. We will show several different constructions of new families of k-rank graphs and C*-algebras based on these complexes, for arbitrary k. The underlying building structure allows explicit computation of the K-theory as well as the explicit spectra computation for the k-graphs.

Fabienne Chouraqui: Connections between the Yang-Baxter equation and Thompson’s group F

22nd June, 2023

The quantum Yang-Baxter equation is an equation in mathematical physics and it lies in the foundation of the theory of quantum groups. One of the fundamental problems is to find all the solutions of this equation. Drinfeld suggested the study of a particular class of solutions, derived from the so-called set-theoretic solutions. A set-theoretic solution of the Yang-Baxter equation is a pair (X,r), where X is a set and

r : X ⨯ X → X ⨯ X     r(x,y)=(σx(y),γy(x))

is a bijective map satisfying r12r23r12 = r23r12r23, where r12 = r ⨯ IdX and r23 = IdX ⨯ r. We define non-degenerate involutive partial solutions as a generalization of non-degenerate involutive set-theoretical solutions of the quantum Yang-Baxter equation (QYBE). The induced operator is not a classical solution of the QYBE, but a braiding operator as in conformal field theory. We define the structure inverse monoid of a non-degenerate involutive partial solution and prove that if the partial solution is square-free, then it embeds into the restricted product of a commutative inverse monoid and an inverse symmetric monoid. Furthermore, we show that there is a connection between partial solutions and the Thompson's group F. This raises the question of whether there are further connections between partial solutions and Thompson's groups in general.

Jeremy Brent Hume: The K-theory of a rational function

22nd June, 2023

The dynamics of iterating a rational function exhibits complicated and interesting behaviour when restricted to points in its Julia set. Kajiwara and Watatani constructed a C*-algebra from a rational function restricted to its Julia set in order to study its dynamics from an operator algebra perspective. They showed the C*-algebras are Kirchberg algebras that satisfy the UCT, and are therefore classified by K-theory. The K-theory groups of these algebras have been computed in some special cases, for instance by Nekrashevych in the case of a hyperbolic and post-critically finite rational function. We compute the K-theory groups for a general rational function using methods different to those used before. In this talk, we discuss our methods and results.