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Raschid Abedin: Classification of D-bialgebras via algebraic geometry

28th October, 2024

In a now classic paper, Belavin and Drinfeld categorized solutions to the classical Yang-Baxter equation (CYBE), an equation crucial to the theory of integrable systems, into three classes: elliptic, trigonometric and rational. It is possible to reproduce this result by geometrizing solutions of the CYBE and then applying algebro-geometric methods. In this talk, we will explain how this approach can be used to categorize Lie bialgebra structures on power series Lie algebras, as well as non-associative generalizations of these structures: D-bialgebra structures on more general power series algebras.

Ilya Chevyrev: Pre-Lie algebras in stochastic PDEs

21st October, 2024

In this talk, I will discuss a general method to renormalize singular stochastic partial differential equations (SPDEs) using the theory of regularity structures. It turns out that, to derive the renormalized equation, one can employ a convenient multi-pre-Lie algebra. The pre-Lie products in this algebra are reminiscent of the pre-Lie product on the Grossman-Larson algebra of trees, but come with several important twists. For the renormalization of SPDEs, the important feature of this multi-pre-Lie algebra is that it is free in a certain sense.

Martin Frankland: Toda brackets in n-angulated categories

17th October, 2024

Geiss, Keller, and Oppermann introduced n-angulated categories to capture the structure found in certain cluster tilting subcategories in quiver representation theory. Jasso and Muro investigated Toda brackets and Massey products in such cluster tilting subcategories by using the ambient triangulated category. In joint work with Sebastian Martensen and Marius Thaule, we introduce Toda brackets in n-angulated categories, generalizing Toda brackets in triangulated categories (the case n = 3). We will look at different constructions of the brackets, their properties, some examples, and some applications.

Kai Meng Tan: Cores and core blocks of Ariki-Koike algebras

15th October, 2024

This talk will consist of two parts. In the first part, we will see how certain results (such as the Nakayama 'Conjecture') for the symmetric groups and Iwahori-Hecke algebras of type A can be generalised to Ariki-Koike algebras using the map from the set of multipartitions to that of (single) partitions first defined by Uglov. In the second part, we look at Fayers's core blocks, and see how these blocks may be classified using the notation of moving vectors first introduced by Yanbo Li and Xiangyu Qi. If time allows, we will discuss Scopes equivalences between these blocks arising as a consequence of this classification

Tom Gannon: Quantization of the universal centralizer and central D-modules

14th October, 2024

We will discuss joint work with Victor Ginzburg that proves a conjecture of Nadler on the existence of a quantization, or non-commutative deformation, of the Knop-Ngô morphism, a morphism of group schemes used in particular by Ngô in his proof of the fundamental lemma in the Langlands programme. We will first explain the representation-theoretic background, give an extended example of this morphism for the group GLn(ℂ), and then present a precise statement of our theorem.

Time permitting, we will also discuss how the tools used to construct this quantization can also be used to prove conjectures of Ben-Zvi and Gunningham, which predict a relationship between the quantization of the Knop-Ngô morphism and the parabolic induction functor, as well as an "exactness" conjecture of Braverman and Kazhdan in the D-module setting.

Slaven Kožić: Representations of the quantum affine vertex algebra associated with the trigonometric R-matrix of A

14th October, 2024

One important problem in the vertex algebra theory is to associate certain vertex algebra-like objects, the quantum vertex algebras, to various classes of quantum groups, such as quantum affine algebras or double Yangians. In this talk, I will discuss this problem in the context of Etingof-Kazhdan's quantum affine vertex algebra Vc(𝔤𝔩N) associated with the trigonometric R-matrix of type A. The main focus will be on the explicit description of the centre of Vc(𝔤𝔩N) at the critical level c = -N and, furthermore, on the connection between certain classes of Vc(𝔤𝔩N)-modules and representation theories of the quantum affine algebra of type A and the orthogonal twisted h-Yangian.

Karlee Westrem: A new symmetric function identity with an application to symmetric group character values

7th October, 2024

Symmetric functions show up in several areas of mathematics including enumerative combinatorics and representation theory. Tewodros Amdeberhan conjectures equalities of Σn character sums over a new set called Ev(λ). When investigating the alternating sum of characters for Ev(λ) written in terms of the inner product of Schur functions and power sum symmetric functions, we found an equality between the alternating sum of power sum symmetric polynomials and a product of monomial symmetric polynomials. As a consequence, a special case of an alternating sum of Σn characters over the set Ev(λ) equals 0.

Ana Agore: Solutions of the set-theoretic Yang-Baxter equation of Frobenius-Separability (FS) type

7th October, 2024

We investigate a special class of solutions of the set-theoretic Yang-Baxter equation, called Frobenius-Separability (FS) type solutions. In particular, we show that the category of solutions of the set-theoretic Yang-Baxter equation of Frobenius-Separability (FS) type is equivalent to the category of pointed Kimura semigroups. As applications, all involutive, idempotent, non-degenerate, surjective, finite order, unitary, or indecomposable solutions of FS type are classified. For instance, if |X| = n, then the number of isomorphism classes of all such solutions on X that are (a) left non-degenerate, (b) bijective, (c) unitary or (d) indecomposable and left-non-degenerate is: (a) the Davis number d(n), (b) Σm|n p(m), where p(m) is the Euler partition number, (c) τ(n) + Σd|n[d/2], where τ is the number of divisors of n, or (d) the Harary number. The automorphism groups of such solutions can also be recovered as automorphism groups Aut(f) of sets X equipped with a single endo-function f : X → X. We describe all groups of the form Aut(f) as iterations of direct and (possibly infinite) wreath products of cyclic or full symmetric groups, characterize the abelian ones as products of cyclic groups, and produce examples of symmetry groups of FS solutions not of the form Aut(f).

Kay Jin Lim: Integral basic algebras

1st October, 2024

Algebras defined over fields of characteristic zero and positive characteristic usually do not behave the same way. In the recent preprint with David J. Benson, we initiate the study by focusing on the integral basic algebras. That is, we consider a p-modular system (K,𝒪,k) and an 𝒪-algebra A where both the algebras K⊗𝒪A and k⊗𝒪A are basic. When the algebra satisfies the right hypotheses, we have equalities of the dimensions of their cohomology groups between simple modules and equalities of graded Cartan numbers. As a case study, we focus on the descent algebras of Coxeter groups. They have been extensively studied since the introduction by Louis Solomon in 1976. We investigate their invariants as mentioned previously, their Ext quivers and representation type. The classification of the representation type in the p = 0 case has previously achieved by Manfred Schocker. In a recent preprint, together with Karin Erdmann, we complete the classification in the p > 0 case.

Ian Tan: Tensor decompositions with applications to LU and SLOCC equivalence of multipartite pure states

30th September, 2024

We introduce a broad lemma, one consequence of which is the higher order singular value decomposition (HOSVD) of tensors defined by DeLathauwer, DeMoor and Vandewalle (2000). By an analogous application of the lemma, we find a complex orthogonal version of the HOSVD. Kraus' (2010) algorithm used the HOSVD to compute normal forms of almost all n-qubit pure states under the action of the local unitary group. Taking advantage of the double cover SL2(ℂ) × SL2(ℂ) → SO4(ℂ), we produce similar algorithms (distinguished by the parity of n) that compute normal forms for almost all n-qubit pure states under the action of the SLOCC group.