During term-time, we hold an Algebra Seminar at 3pm on Tuesdays.
In this page, you can find all the information on the seminar (currently organised by Francesca Fedele, Ilaria Colazzo and Peter Huston) since September 2025.
Unless otherwise specified, Algebra seminars take place during term time on Tuesdays at 3.00pm in the MALL, School of Mathematics, University of Leeds.
While an affine algebraic group is a subgroup of the general linear group that is defined by some polynomial equations, an affine difference algebraic group is a subgroup of the general linear group that is defined by difference polynomial equations. In this talk I will introduce difference algebra and difference algebraic geometry, before explaining how we can use difference algebraic groups to find a class of difference ideals that are finitely difference generated.
Long before having taken their name, thus earned their distinguished place in an area in the intersection of algebra, Knots Theory and combinatorics now a days, Fully Commutative Elements served in the shadows for years, being used as element of basis of the known Temperley-Lieb algebras. It turns out that "they" have their own story. I shall enter by the pure algebra door by giving the general definition, which is simple "by definition". It is a fact that 55 minutes would not be enough to tell the importance of such a class of elements, yet I hope having enough time to talk about the present rather than the past and and so to present some interesting open problems. Please enjoy.
I will talk about a joint work with Tyrone Crisp and Uri Onn about the complex representation theory of the groups GL_n(Z/p^r). Following the work of Zelevinsky, we know that when r=1 the complex representation theory of the groups GL_n(Z/p) decomposes nicely into a combinatorial part and an arithmetic part. The representation theory of GL_n(Z/p^r) where r>1 seems to be much more complicated. We conjecture that a similar decomposition exists there. In this talk I will explain how we tackle this conjecture, using tools from symmetric monoidal categories and from classical representation theory. The main insight from symmetric monoidal categories is that the languages of finite groups and finite rings intertwine here, and that studying the representation theory of GL_n(Z/p^r) naturally contains the study of groups of the form Aut_R(M) for general finite rings R and finite modules M. This much more general framework, together with classical tools from representation theory, gives much better insight into the conjecture.
In a 2019 paper, Kadar, Martin, and Yu, introduced the notion of left height for a Brauer diagram, and used it to define a family of subalgebras of the Brauer algebra which interpolate between the Temperley-Lieb (sub)algebra, and the full Brauer Algebra. Early calculations indicated that for intermediate height values, these algebras exhibited novel semi-simplicity criteria distinct from the classical criteria in the Temperley-Lieb case (roots of unity), and Brauer case (integers). In this talk, we present ongoing work aimed characterising these criteria; rules for computing Gram-matrix determinants for all standard modules are given in some low height cases, leading to a general conjecture. This conjecture involves introducing a Chebyshev series of polynomials for each integer partition, the roots of which seem to “resemble” roots of unity for large n. Time permitting, we will discuss some of the representation theoretic consequences of our calculations for the Kadar-Yu algebras in low rank.
Finding Morita invariants of fusion categories enriched over a fixed modular tensor category is important because, by a form of the cobordism hypothesis, they correspond (2+1)D twice-extended framed TQFTs, and therefore to physical characteristics of topological phases of matter. In this talk, we will introduce the notions of fusion 1- and 2-category and see how they appear naturally in higher linear algebra. We will then see how the Morita 3-category of fusion categories, bimodule categories, bimodule functors, and natural transformations can be equivalently described by concrete Morita invariants, so that choices of representatives of Morita class need not be made.
I start with the definition of cluster integrable systems a la Goncharov and Kenyon, defined by convex Newton polygons, up to the action of $SA(2,\mathbb{Z})$. There are several arguments requiring that to complete the picture, this class should be extended by their Hamiltonian reductions, which can be performed preserving the structure of cluster variety.
Quantum toroidal algebras Uq(g_tor) are the ‘double affine’ objects within the quantum setting. After reviewing and motivating the more well-studied finite and affine quantum groups, we’ll introduce these quantum toroidal algebras and outline a collection of new results.
First we shall construct an action of the double affine braid group, and derive anti-involutions ψ of Uq(g_tor) that form part of a surprising modular GL(2,Z) symmetry.
Then, using ψ, we will investigate the representation theory of Uq(g_tor), obtaining well-defined tensor products and intertwiners (R-matrices) for an important category of modules.
Profinite groups are a class of topological groups with close ties to finite groups. Many of the questions around group rings of finite groups and their representation theory have profinite analogues. I will report on results that generalise block fusion systems and Puig's theory of nilpotent blocks to the profinite setting, and a resulting description of blocks with infinite dihedral defect groups. This is joint work with MacQuarrie and Franquiz Flores.
A torsor, also known as a principal homogeneous space, is an algebraic variety with a simply transitive action of an algebraic group. These play a main role in the arithmetic theory of algebraic groups and an important question is to find conditions that guarantee that the torsor is trivial, i.e., isomorphic to the algebraic group itself with action given by left multiplication. In this talk we will discuss analogous questions in the realm of difference algebraic geometry, i.e., instead of just algebraic equations, we allow difference algebraic equations in the definition of our groups and varieties.
This is joint work with Annette Bachmayr.
On the space of matrices with rational (trigonometric/elliptic) entries there is a well-known Lie-Poisson structure, the ``r-matrix structure’’. It is an essential structure underlying the Hamiltonian dynamics of the vast majority of integrable systems, isospectral and isomonodromic evolution equations. The known r-matrices depend on parameter in rational way (trig/elliptic, respectively) and hence we think of them on the Riemann sphere (cylinder/torus).
In a relatively abstract Hamiltonian framework the isospectral evolution equations are generalized to higher genus Riemann surfaces as the “Hitchin systems”, an evolutionary integrable system on the moduli space of vector bundles. On the isomonodromic side main progress is attributable to Krichever who used a quite explicit coordinatization of vector bundles on Riemann surfaces that we can call “Tyurin parametrization”.
In this talk I report on the fully explicit generalization of the r-matrix structure to an arbitrary genus Riemann surface merging the Tyurin-Krichever approach with the general framework of Hitchin’s. The key tool is a (fully explicit) matrix-valued kernel that plays crucial role also in setting up integral equations in related area of the "non-abelian steepest descent” method.
