Julia Plavnik: Remarks on the tensor product property for support varieties for finite tensor categories

22nd April, 2024

For non-semisimple tensor categories satisfying some finiteness conditions, support varieties are meaningful geometric invariants of objects. Their theory began in the work of Quillen and Carlson on finite group representations. In more recent years, the theory of support varieties was generalized in many directions, including representations of finite-dimensional Hopf algebras and self-injective algebras, and objects in finite tensor categories and triangulated categories, among others.

In this talk, we will start by introducing the definition of support varieties for finite tensor categories and some of their basic properties. We will also present some conditions under which the tensor product property holds for support varieties, and we will present some applications to certain Hopf algebras. We will also discuss a construction of non-semisimple finite tensor categories with finitely generated cohomology for which the tensor product property does not hold for support varieties.

Kang Lu: A Drinfeld presentation of twisted Yangians via degeneration

2nd March, 2024

We formulate a new family of algebras, twisted Yangians (of split type) in current generators and relations, via degeneration of Drinfeld presentations of affine iquantum groups (associated with split Satake diagrams). These new algebras admit PBW type bases and are shown to be a deformation of twisted current algebras. For type AI, it matches with the Drinfeld presentation of twisted Yangian obtained via Gauss decomposition. We conjecture that our twisted Yangians are isomorphic to twisted Yangians constructed in RTT presentation.

Oleksandr Tsymbaliuk: Lyndon words and fused currents in shuffle algebra

2nd March, 2024

Classical q-shuffle algebras provide combinatorial models for the positive half Uq(𝔫) of a finite quantum group. We define a loop version of this construction, yielding a combinatorial model for the positive half Uq(L𝔫) of a quantum loop group. In particular, we construct a PBW basis of Uq(L𝔫) indexed by standard Lyndon words, generalizing the work of Lalonde-Ram, Leclerc and Rosso in the Uq(𝔫) case. We also connect this to Enriquez's degeneration A of the elliptic algebras of Feigin-Odesskii, proving a conjecture that describes the image of the embedding Uq(L𝔫)→A in terms of pole and wheel conditions. The talk shall conclude with the shuffle interpretations of fused currents proposed by Ding-Khoroshkin.

Thomas Haines: Pavings of convolution fibres and applications

1st March, 2024

A convolution morphism is the geometric analogue of the convolution of functions in a Hecke algebra. The properties of fibres of convolution morphisms are used in a variety of ways in the geometric Langlands programme and in the study of Schubert varieties. I will explain a very general result about cellular pavings of fibres of convolution morphisms in the setting of partial affine flag varieties, as well as applications related to the very purity and parity vanishing of cohomology of Schubert varieties over finite fields, structure constants for parahoric Hecke algebras, and the (motivic) geometric Satake equivalence.

Ziqing Xiang: Quantum Wreath Products and Their Representations

19th February, 2024

We introduce a new notion called the quantum wreath product, which produces an algebra B ≀Q H(d) from a given associative algebra B, a positive integer d, and a choice Q = (R, S, ρ, σ) of parameters. Important examples include many variants of the Hecke algebras, such as the Ariki-Koike algebras, the affine Hecke algebras and their degenerate version, Wan-Wang’s wreath Hecke algebras, Rosso-Savage’s (affine) Frobenius Hecke algebras, Kleshchev-Muth’s affine zigzag algebras, and the Hu algebra that quantizes the wreath product Σm ≀ Σ2 between symmetric groups. We will discuss the bases of quantum wreath product algebras, and some of their representations.

Lars Winther Christensen: The derived category of a regular ring

25th January, 2024

Recall that a noetherian ring R is regular if every finitely generated R-module has finite projective dimension. In a paper from 2009, Iacob and Iyengar characterize the regularity of R in terms of properties of (unbounded) R-complexes. Their proofs build on results of Jorgensen, Krause, and Neeman on compact generation of the homotopy categories of complexes of projective/injective/flat modules. In the commutative case, these results can be obtained with derived category methods in local algebra. I will illustrate how this is done by proving that the following conditions are equivalent for a commutative noetherian ring R:

1) R is regular.

2) Every complex of finitely generated projective R-modules is semi-projective.

3) Every complex of projective R-modules is semi-projective.

4) Every acyclic complex of projective R-modules is contractible.

The second condition is new, compared to the 2009 results, and relating it to the regularity of R is the novel part of the proof. This argument also plays a central role in the new proof of the corresponding results for complexes of injective modules and complexes of flat modules.

Alexander Wilson: Super Multiset RSK and a Mixed Multiset Partition Algebra

22nd January, 2024

Through dualities on representations on tensor powers and symmetric powers respectively, the partition algebra and multiset partition algebra have been used to study long-standing questions in the representation theory of the symmetric group. These algebras enjoy distinguished bases whose product can be described on graph-theoretic diagrams. We extend this story to exterior powers, leading to the introduction of the mixed multiset partition algebra and a generalization of RSK that links the algebra’s graph-theoretic basis to a tableau basis for its irreducible representations.

Sergio López-Permouth: Basic Extension Modules (All bases are created equal, but some are more equal than others)

8th January, 2024

We report on ongoing research about a module-theoretic construction which, when successful, yields natural extensions of infinite-dimensional modules over arbitrary algebras. Whether the construction works or not depends on the basis that one chooses to carry on such a construction. Bases that work are said to be amenable. A natural example on which one may focus is when the module is the algebra itself. For instance, a great deal of the work done so far has focused on the infinite-dimensional algebra of polynomials on a single variable. We will see that amenability and related notions serve to classify the distinct bases according to interesting complementary properties having to do with the types of relations induced on them by the properties of their change-of-basis matrices.

Jason Gaddis: Rigidity of quadratic Poisson algebras

18th December, 2023

The Shephard-Todd-Chevalley Theorem gives conditions for the invariant ring of a polynomial ring to again be polynomial. However, this behaviour is rarely observed for non-commutative algebras. For example, the invariant ring of the first Weyl algebra by a finite group is not isomorphic to the first Weyl algebra. In this talk, I will discuss this rigidity in the context of quadratic Poisson algebras. A key example will be those Poisson polynomial algebras with skew-symmetric structure.

Askar Dzhumadil’daev: Rota-Baxter algebras with non-zero weights

18th December, 2023

For an associative commutative algebra A with Rota-Baxter operator R : A → A with weight λ denote by AR an algebra with linear space A and multiplication a ◦ b = aR(b). Let AR− and AR+ be the algebra AR under Lie and Jordan commutators. If λ = 0, then the algebra AR = (A, ◦) is Zinbiel, AR+ is associative, and AR− is Tortkara. We find polynomial identities of algebras AR, AR− and AR+ in the case λ ≠ 0. We prove that AR− is Tortkara. AR+ satisfies an identity of degree 5. In the case λ ≠ 0, the algebra AR is not associative-admissible.