Tag - Representations of algebras

Kaveh Mousavand: Some applications of bricks in classical and modern problems in representation theory

12th December, 2023

Bricks (also known as Schur representations) form a special subfamily of indecomposable modules, and they are used in the algebraic and geometric study of representation theory of algebras. We start by looking at some classical results on bricks, including a characterization of locally representation-directed algebras (due to Dräxler). Then, we consider some new directions of research in which bricks have played crucial roles. More specifically, we briefly recall an elegant correspondence between bricks and indecomposable τ-rigid-modules (due to Demonet-Iyama-Jasso), which has many applications in τ-tilting theory. We use the notion of τ-rigidity to give a new characterization of locally representation-directed algebras, and to further generalize this family. If time permits, we also report on some new results on an open conjecture (so-called the 2nd brick-Brauer-Thrall conjecture) which I posed in 2019.

Vasily Krylov: On Hikita-Nakajima conjecture for quiver varieties and Slodowy slices

22nd March, 2023

Symplectic duality predicts that symplectic singularities should come in pairs. For example, Nakajima quiver varieties are conjecturally dual to BFN Coulomb branches (of the corresponding quiver theories). Another family of potentially symplectically dual pairs was described recently in the works of Losev, Mason-Brown, and Matvieievskyi: they describe symplectically duals to Słodowy slices to nilpotent orbits.

In this talk, we will discuss the Hikita-Nakajima conjecture that relates the geometry of symplectically dual varieties. It turns out that the conjecture is very likely to hold for quiver varieties (as was predicted by Nakajima) but does not quite hold for Słodowy slices and arbitrary Higgs branches. We will explain certain simplification of this conjecture that may work in general. We will discuss a possible approach toward the proof of this conjecture. The approach is highly based on the ideas of Bellamy, Braverman, Kamnitzer, Losev, Tingley, Webster, Weekes, Yacobi, and their co-authors.

We will illustrate the approach on the examples of ADHM space (for which Hikita-Nakajima conjecture is true as stated) and for certain Słodowy varieties.

Sergio Lopez-Permouth: On the isomorphism problem for basic modules

30th June, 2022

While mutual congeniality of bases has been known to guarantee that basic modules from so related bases are isomorphic, the question of what can be said about isomorphism of basic modules in general has remained open.

We show that neither of two possible extremes need hold. For some algebras it is possible for basic modules to be non-isomorphic. Also, it is possible, for some algebras, that all basic modules are isomorphic. We show that there are at least as many pairwise non-isomorphic basic modules over the F-algebra F[x] of polynomials in a single variable as there are elements in F. We show that basic modules over F[x] can be non-isomorphic when they are induced by discordant bases and also even when there is a (non-mutual) congeniality among them. In the process and as a byproduct, we introduce the notion of domains of divisibility of modules over arbitrary rings and explore some of the properties of a divisibility profile.

At the opposite end of the spectrum, we present an algebra where all basic modules are isomorphic, regardless of congeniality.

Yuriy Drozd: Morita Theory for non-commutative varieties

25th March, 2022

Morita theorem gives a criterion of equivalence of categories of modules over rings. On the other hand, Gabriel proved that the category of coherent sheaves defines a Noetherian scheme up to isomorphism. We have established a result which is in a sense, a union and a combination of these two theorems. Namely, we show that the category of coherent sheaves over a Noetherian non-commutative scheme completely defines its centre and the schemes with the same centre are Morita equivalent if and only if one of them is isomorphic to the scheme of endomorphisms of a local progeneretor of the other.

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.

Claude Cibils: Controlling the global dimension

10th December, 2020

The global dimension of an associative algebra A over a a field is a measure of the complexity of its representations. It is 0 if A is a matrix algebra. It is 1 if A is a path algebras of quivers without directed cycles. It is infinite if A is the algebra of dual numbers.

I will give a brief introduction to Hochschild homology (1945), in order to explain Han's conjecture (2006): for finite-dimensional algebras, the Hochschild homology should control the finiteness of the global dimension.

Next, I will present some progress made in showing Han's conjecture, using the relative version of Hochschild homology (1956) with respect to a subalgebra B. This theory was little used until recently. Now we have a Jacobi-Zariski long nearly exact sequence which relates the usual and relative versions of Hochschild homology. Its gap to be exact is approximated by a spectral sequence which has Tor functors in its first page, of B-tensor powers of A/B. This tool enables to show, for instance, that the class of algebras verifying Han's conjecture is closed by bounded extensions of algebras.

James East: Presentations for tensor categories

15th May, 2020

Many well-known families of groups and semigroups have natural categorical analogues: e.g., full transformation categories, symmetric inverse categories, as well as categories of partitions, Brauer/Temperley-Lieb diagrams, braids and vines. This talk discusses presentations (by generators and relations) for such categories, utilising additional tensor/monoidal operations. The methods are quite general, and apply to a wide class of (strict) tensor categories with one-sided units.