# Symposium on Koszul Algebras and Abelian Categories

Wednesday 11 May 2022 14:15–17:00 Aud. D1 (1531-113)
Symposium

The (extended) ADE Coxeter-Dynkin diagrams have a knack for showing up in various finite classification problems in mathematics, e.g. classifying representation finite (tame) hereditary algebras. For simple undirected graphs, they solve the problem of characterizing the simple undirected graphs with largest adjacency eigenvalue less than (or equal to) two. By work of A’Campo (1976), it is known that this is no accident.

As it turns out, both classification problems have natural higher dimensional generalizations, and I discuss some connections analogous to those shown by A’Campo. In particular, I cover recent progress on classifying quadratic monomial 2-representation finite algebras via spectral methods, connections with bipartite reflexive graphs, and an associated Diophantine equation.

15:15-16:00 Job D. Rock (Ghent): Composition Series of Arbitrary Cardinality in Abelian Categories

We generalize composition series in an abelian category to allow the multiset of composition factors to have arbitrary cardinality. Motivating examples include pointwise finite-dimensional persistence modules, Prüfer modules, and presheaves, where we generalize the notion of support used in quiver representations. A “Jordan–Hölder–Schreier” like theorem holds when objects satisfy a set of four axioms. With one additional axiom, we can prove a result that generalizes some properties of a length category. This is based on joint work (arXiv:2106.01868) with Eric J. Hanson.

16:15-17:00 Paul Smith (Washington): An overview of elliptic algebras

Elliptic algebras form a family of connected graded algebras defined over the complex numbers that depend on a pair of relatively prime positive integers $n'>k$ and a complex elliptic curve, $E$, and a point on it. They were defined in this generality by Feigin and Odesskii in 1989 and have not been studied much since then, in part because Feigin and Odesskii's papers were long on assertions and short on proofs. In joint work with Alex Chirvasitu and Ryo Kanda over the past 5 years we have proved a number of results (about 250 journal pages) about them that have put the subject on a firmer footing. For fixed $(n,k,E)$ they form a flat family of deformations of the polynomial ring on $n$ variables and share many homological properties with that polynomial ring: for example, they are Koszul and have global homological dimension $n$. Their quadratic duals are finite dimensional algebras, deformations of exterior algebras, therefore very wild, and (because they are connected) not rich in combinatorial properties. However, they are very rich in geometric connections the relevant objects being varieties built from elliptic curves. Their definition is via generators and relations involving theta functions. This means they are complicated. In particular, they have no explicit nice basis so calculations with elements are essentially impossible.

I assume no prior knowledge of elliptic algebras.

Organised by: AarHomAlg
Contact: Karin M. Jacobsen Revised: 11.05.2022