Tag - Associative rings

Agata Smoktunowicz: Some questions related to nilpotent rings and braces

27th February, 2021

In around 2005, Wolfgang Rump introduced braces, a generalization of nilpotent rings to describe all involutive, non-degenerate set-theoretic solutions of the Yang-Baxter equation. This formulation then rapidly found application in other research areas. This talk will review these applications.

Definition. A set A with binary operations of addition +, and multiplication ∘ is a brace if (A, +) is an abelian group, (A, ∘) is a group and a ∘ (b+c)+a = a ∘ b+a ∘ c for every a, b, c ∈ A. It follows from this definition that every nilpotent ring with the usual addition and with multiplication a ∘ b = ab + a + b is a brace.

Braces have been shown to be equivalent to several concepts in group theory such as groups with bijective 1-cocycles and regular subgroups of the holomorph of abelian groups. In algebraic number theory there is a correspondence between braces and Hopf-Galois extensions of abelian type first observed by David Bachiller. There is also connection between R-braces and pre-Lie algebras discovered by Wolfgang Rump in 2014. One generator braces have been shown to describe indecomposable, involutive solutions of the Yang-Baxter equation.

On the other hand, Anastasia Doikou and Robert Weston have recently discovered some fascinating connections between braces and quantum integrable systems. In particular, to find solutions of the set-theoretic reflection equation it is needed to solve problems on some polynomial identities in nilpotent rings. Because previously the theory of polynomial identities was mainly developed for prime rings, and for the reflection equation we only consider nilpotent rings, there are no known methods for solving such problems. We will mention some open problems on polynomial identities in nilpotent rings which appear in this situation.

Giles Gardam: Kaplansky’s conjectures

22nd February, 2021

Kaplansky made various related conjectures about group rings, especially for torsion-free groups. For example, the zero divisors conjecture predicts that if K is a field and G is a torsion-free group, then the group ring K[G] has no zero divisors. I will survey what is known about the conjectures, including their relationships to each other and to other group properties such as orderability, and present some recent progress.

Alistair Savage: Affinization of monoidal categories

2nd February, 2021

We define the affinization of an arbitrary monoidal category, corresponding to the category of string diagrams on the cylinder. We also give an alternative characterization in terms of adjoining dot generators to the category. The affinization formalizes and unifies many constructions appearing in the literature. We describe a large number of examples coming from Hecke-type algebras, braids, tangles, and knot invariants.

Nicolas Jacon: Cores of Ariki-Koike algebras

8th December, 2020

We study a natural generalization of the notion of cores for l-partitions: the (e,s)-cores. We relate this notion with the notion of weight as defined by Fayers and use it to describe the blocks of Ariki-Koike algebras.

Laura Rider: Modular Perverse Sheaves on the Affine Flag Variety

7th December, 2020

There are two categorical realizations of the affine Hecke algebra: constructible sheaves on the affine flag variety and coherent sheaves on the Langlands dual Steinberg variety. A fundamental problem in geometric representation theory is to relate these two categories by a category equivalence. This was achieved by Bezrukavnikov in characteristic 0 about a decade ago. In this talk, I will discuss a first step toward solving this problem in the modular case joint with R. Bezrukavnikov and S. Riche.

Ualbai Umirbaev: Automorphism groups of free algebras

4th December, 2020

There are many interesting results on the structure of the automorphism group Aut(Fn) and the outer automorphism group Out(Fn) of the free group Fn of rank n. Unfortunately, the theory of automorphism groups of free algebras over a field is not very rich and many problems are still open. I will describe some results and recall some open questions on the structures of the automorphism groups of:

   1.  the polynomial algebra K[x1,x2,…,xn] of rank n over a field K;
   2.  the free associative algebra K<x1,x2,...,xn> of rank n over K; and
   3.  the free Lie algebra Lie<x1,x2,...,xn> of rank n over K.

Agatha Atkarskaya: Small cancellation rings

4th December, 2020

The theory of small cancellation groups is well known. In this paper we introduce the notion of Group-like Small Cancellation Ring. This is the main result of the paper. We define this ring axiomatically, by generators and defining relations. The relations must satisfy three types of axioms. The major one among them is called the Small Cancellation Axiom. We show that the obtained ring is non-trivial. Moreover, we show that this ring enjoys a global filtration that agrees with relations, find a basis of the ring as a vector space and establish the corresponding structure theorems. It turns out that the defined ring possesses a kind of Gröbner basis and a greedy algorithm. Finally, this ring can be used as a first step towards the iterated small cancellation theory which hopefully plays a similar role in constructing examples of rings with exotic properties as small cancellation groups do in group theory. Joint results with A. Kanel-Belov, E. Plotkin, E. Rips.

Eli Aljadeff: Generic Azumaya G-graded algebras

26th November, 2020

The algebra of generic nxn-matrices and its localizations (e.g. the generic division algebra) has attracted much attention among researchers in different areas as PI theory, Brauer theory and algebraic geometry. We construct the corresponding generic objects for an arbitrary finite dimensional G-graded simple algebra where G is a finite group. In particular we construct a generic G-graded Azumaya algebra which represents all forms in the sense of descent theory of a finite dimensional G-graded simple algebra.

Qi Wang: On τ-tilting finiteness of Schur algebras

17th November, 2020

Support τ-tilting modules are introduced by Adachi, Iyama and Reiten in 2012 as a generalization of classical tilting modules. One of the importance of these modules is that they are bijectively corresponding to many other objects, such as two-term silting complexes and left finite semibricks. Let V be an n-dimensional vector space over an algebraically closed field 𝔽 of characteristic p. Then, the Schur algebra S(n,r) is defined as the endomorphism ring End𝔽Gr(V⊗r) over the group algebra 𝔽Gr of the symmetric group Gr. In this talk, we discuss when the Schur algebra S(n,r) has only finitely many pairwise non-isomorphic basic support τ-tilting modules.