Tag - Associative rings

Max Gurevich: New constructions for irreducible representations in monoidal categories of type A

28th May, 2021

One ever-recurring goal of Lie theory is the quest for effective and elegant descriptions of collections of simple objects in categories of interest. A cornerstone feat achieved by Zelevinsky in that regard, was the combinatorial explication of the Langlands classification for smooth irreducible representations of p-adic GLn. It was a forerunner for an exploration of similar classifications for various categories of similar nature, such as modules over affine Hecke algebras or quantum affine algebras, to name a few. A next step - reaching an effective understanding of all reducible finite-length representations remains largely a difficult task throughout these settings.

Recently, joint with Erez Lapid, we have revisited the original Zelevinsky setting by suggesting a refined construction of all irreducible representations, with the hope of shedding light on standing decomposition problems. This construction applies the Robinson-Schensted-Knuth transform, while categorifying the determinantal Doubilet-Rota-Stein basis for matrix polynomial rings appearing in invariant theory. In this talk, I would like to introduce the new construction into the setting of modules over quiver Hecke (KLR) algebras. In type A, this category may be viewed as a quantization/gradation of the category of representations of p-adic groups. I will explain how adopting that point of view and exploiting recent developments in the subject (such as the normal sequence notion of Kashiwara-Kim) brings some conjectural properties of the RSK construction (back in the p-adic setting) into resolution. Time permits, I will discuss the relevance of the RSK construction to the representation theory of cyclotomic Hecke algebras.

Alistair Savage: Affine Hecke algebras and the elliptic Hall algebra

13th May, 2021

The elliptic Hall algebra has appeared in many different contexts in representation theory and geometry under different names. We will explain how this algebra is categorified by the quantum Heisenberg category, which is a diagrammatic category modelled on affine Hecke algebras. This categorification can be used to construct large families of representations for the elliptic Hall algebra.

Benjamin Fine: Elementary and universal theories of group rings

6th May, 2021

In a series of papers the above authors examined the relationship between the universal and elementary theory of a group ring R[G] and the corresponding universal and elementary theory of the associated group G and ring R. Here we assume that R is a commutative ring with identity 1 ≠ 0. Of course, these are relative to an appropriate logical language L0, L1, L2 for groups, rings and group rings respectively. Axiom systems for these were provided. Kharlampovich and Myasnikov, as part of the proof of the Tarskii theorems, prove that the elementary theory of free groups is decidable. For a group ring they have proved that the first-order theory (in the language of ring theory) is not decidable and have studied equations over group rings, especially for torsion-free hyperbolic groups. We examined and survey extensions of Tarksi-like results to the collection of group rings and examine relationships between the universal and elementary theories of the corresponding groups and rings and the corresponding universal theory of the formed group ring. To accomplish this we introduce different first-order languages with equality whose model classes are respectively groups, rings and group rings. We prove that if R[G] is elementarily equivalent to S[H] then simultaneously the group G is elementarily equivalent to the group H and the ring R is elementarily equivalent to the ring S with respect to the appropriate languages. Further if G is universally equivalent to a nonabelian free group F and R is universally equivalent to the integers ℤ then R[G] is universally equivalent to ℤ[F] again with respect to an appropriate language. It was proved that if R[G] is elementarily equivalent to S[H] with respect to L2, then simultaneously the group G is elementarily equivalent to the group H with respect to L0, and the ring R is elementarily equivalent to the ring S with respect to L1.

The structure of group rings is related to the Kaplansky zero-divisor conjecture. A Kaplansky group is a torsion-free gorup which satisfies the Kaplansky conjecture. We next show that each of the classes of left-orderable groups and orderable groups is a quasivariety with undecidable theory. In the case of orderable groups, we find an explicit set of universal axioms. We have that 𝒦 the class of Kaplansky groups is the model class of a set of universal sentences in the language of group theory. We also give a characterization of when two groups in 𝒦 or more generally two torsion-free groups are universally equivalent.

Finally we consider F to be a rank 2 free group and ℤ be the ring of integers. we call ℤ[F] a free group ring. Examining the universal theory of the free group ring ℤ[F] the hazy conjecture was made that the universal sentences true in ℤ[F] are precisely the universal sentences true in F modified appropriately for group ring theory and the converse that the universal sentences true in F are the universal sentences true in ℤ[F] modified appropriately for group theory. We prove that this conjecture is true in terms of axiom systems for ℤ[F].

Julia Plavnik: Tensor product and support varieties for finite tensor categories

3rd May, 2021

In this talk, we will consider the support variety theory for Hopf algebras and finite tensor categories. We will start by presenting the basic definitions and properties of these algebraic structures (Hopf algebras and tensor categories) and then we will introduce the theory of support varieties for them. We will analyze questions about projectivity and tensor products by using support theory and we will illustrate this via some examples. If time allows, we will discuss some deeper results involving complexity, realization, and connectedness of the varieties.

Daniele Rosso: Fixed rings of twisted generalized Weyl algebras

12th April, 2021

Twisted generalized Weyl algebras (TGWAs) are a large family of algebras that includes several algebras of interest for ring theory and representation theory, like Weyl algebras and quotients of the enveloping algebra of 𝔰𝔩2. In this work, we study invariants of TGWAs under diagonal graded automorphisms. Under certain conditions, we are able to show that the fixed ring of a TGWA by such an automorphism is again a TGWA. We apply this theorem to study properties of the fixed ring, such as the Noetherian property and simplicity. We also look at the behavior of simple weight modules for TGWAs when restricted to the action of the fixed ring.

Catharina Stroppel: Verlinde rings and DAHA actions

16th March, 2021

In this talk we will briefly recall how quantum groups at roots give rise Verlinde algebras which can be realised as Grothendieck rings of certain monoidal categories. The ring structure is quite interesting and was very much studied in type A. I will try to explain how one gets a natural action of certain double affine Hecke algebras and show how known properties of these rings can be deduced from this action and in which sense modularity of the tensor category is encoded.

Giles Gardam: Kaplansky’s conjectures

11th March, 2021

Three conjectures on group rings of torsion-free groups are commonly attributed to Kaplansky, namely the unit, zero divisor and idempotent conjectures. For example, the zero divisor 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 conjectures and group properties, and finish with my recent counterexample to the unit conjecture.

Vladimir Dotsenko: Diamond Lemma and the Maurer-Cartan equation

5th March, 2021

I shall outline a new approach to the Composition-Diamond Lemma for rewriting systems/Gröbner-Shirshov bases; more specifically, I shall explain how the Maurer-Cartan equation in the tangent complex of a monomial algebra leads to many different versions of the Composition-Diamond Lemma, one for each representative of the tangent complex arising from a multigraded resolution of such algebra.

Aaron Yi Rui Low: Adjustment matrices

2nd March, 2021

James's Conjecture predicts that the adjustment matrix for weight w blocks of the Iwahori-Hecke algebras ℋn and the q-Schur algebras 𝒮n is the identity matrix when w < char(F). Fayers has proved James's Conjecture for blocks of ℋn of weights 3 and 4. We shall discuss some results on adjustment matrices that have been used to prove James's Conjecture for blocks of 𝒮n of weights 3 and 4 in an upcoming paper. If time permits, we will look at a proof of the weight 3 case.