Seminars in Non-Associative Rings

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David Fernández: Non-commutative Poisson geometry and pre-Calabi-Yau algebras

2nd December, 2024

In order to define suitable non-commutative Poisson structures, M. Van den Bergh introduced double Poisson algebras and double quasi-Poisson algebras. Furthermore, N. Iyudu and M. Kontsevich found an insightful correspondence between double Poisson algebras and pre-Calabi-Yau algebras; certain cyclic A∞-algebras which can be seen as non-commutative versions of shifted Poisson manifolds. In this talk, I will present an extension of the Iyudu-Kontsevich correspondence to the differential graded setting. I will also explain how double quasi-Poisson algebras give rise to pre-Calabi-Yau algebras.

Jon Woolf: Heart fans as universal phase diagrams

28th November, 2024

The heart fan is a new convex-geometric invariant of an abelian category which captures interesting aspects of the related homological algebra. I will review the construction and some of its key properties, illustrating them through examples. In particular, I will explain how the heart fan can be viewed as a 'universal phase diagram' for Bridgeland stability conditions with the given heart.

Carlos André: Supercharacters of adjoint groups of radical rings and related subgroups

25th November, 2024

Describing the conjugacy classes and/or irreducible characters of the unitriangular group over a finite field is known to be an impossibly difficult problem. Superclasses and supercharacters have been introduced (under the names of "basic varieties" and "basic characters") as an attempt to approximate conjugacy classes and irreducible characters using a cruder version of Kirillov's method of coadjoint orbits. In the past thirty years, these notions have been recognized in several areas (seemingly unrelated to representation theory): exponential sums in number theory, random walks in probability and statistics, association schemes in algebraic combinatorics... In this talk, we will describe and illustrate the main ideas and recent developments of the standard supercharacter theory of adjoint groups of radical rings. We will explore the close relation to Schur rings, and extend a well-known factorization of supercharacters of unitriangular groups which explains the alternative definition as basic characters.

Vladimir Sosnilo: Resolutions of c-categories

21st November, 2024

Many results about schemes can be generalized to the non-commutative setting of stable ∞-categories. For bounded weighted categories, things are even better: one can formulate and prove a natural analogue of the theorem of Dundas-Goodwillie-McCarthy which is one of the fundamental tools in studying algebraic K-theory. However, in algebraic geometry bounded weighted categories do not show up very often: for instance, the ∞-category of perfect complexes over a scheme X only admits a reasonable weight structure when X is affine. We introduce a new notion of a c-category which is designed to cover a diverse class of geometric examples, including all quasi-compact quasi-separated schemes, yet allowing for all the weighted arguments to work out in this setting. Our main result shows that a c-category can be resolved in finitely many steps by categories of perfect complexes over connective ring spectra. This allows us to prove an analogue of the DGM theorem for c-categories, as well as the vanishing of their Hochschild homology below a certain degree. We also show that either of the following admits a structure of a c-category:

1. the derived category of any exact (∞-)category that has finite Ext-dimension;

2. the subcategory of compact objects in any weakly approximable stable ∞-category in the sense of Neeman.

In particular, using the computation of the Hochschild homology for c-categories mentioned before we obtain that the category of module-spectra over the ring C*(𝕊2) of cochains over the 2-sphere is not weakly approximable.

Pablo Zadunaisky: Clebsch-Gordan revisited

19th November, 2024

By an ultra classical result, the tensor product of a simple representation of 𝔤𝔩n(ℂ) and its defining representation decomposes as a direct sum of simple representations without multiplicities. This means that for each highest weight, the space of highest weight vectors is 1-dimensional. We will give an explicit construction of these highest weight vectors, and show that they arise from the action of certain elements in the enveloping algebra of 𝔤𝔩n(ℂ) + 𝔤𝔩n(ℂ) on the tensor product. These elements are independent of the simple representation we started with, and in fact produce highest weight vectors in several other contexts.

Emanuele Macrì: Deformations of t-structures

14th November, 2024

Bridgeland stability conditions were introduced about 20 years ago, with motivations from algebraic geometry, representation theory, and physics. One of the fundamental problems is that we currently lack methods to construct and study such stability conditions in full generality. In this talk, I will present a new technique to construct stability conditions by deformations, based on joint works with Li, Perry, Stellari, and Zhao. As an application, we can construct stability conditions on very general abelian varieties and deformations of Hilbert schemes of points on K3 surfaces, and we prove a conjecture by Kuznetsov and Shinder on quartic double solids.

Carla Rizzo: Differential identities, almost polynomial growth and matrix algebras

11th November, 2024

Let F be a field of characteristic zero, L a Lie algebra over F, and A an L-algebra - that is, an associative algebra over F with an action of L induced by derivations. This action of L on A can be extended to an action of its universal enveloping algebra U(L), leading to the concept of L-identities or differential identities of A: polynomials in variables xu:= u(x), where u ∈ U(L), that vanish under all substitutions of elements from A. Differential identities were first introduced by Kharchenko in 1978, and, in later years, subsequent work by Gordienko and Kochetov has spurred a renewed interest in both their structure and quantitative properties. In this talk, I will present recent results on the differential identities of matrix L-algebras, with a particular focus on their classification and growth behaviour.

David Jorgensen: Asymptotic vanishing of cohomology in triangulated categories

7th November, 2024

Given a graded-commutative ring acting centrally on a triangulated category, the main result of this talk shows that if the cohomology of a pair of objects of the triangulated category is finitely generated over the ring acting centrally, then the asymptotic vanishing of the cohomology is well-behaved. In particular, enough consecutive asymptotic vanishing of cohomology implies all eventual vanishing. Several key applications are also given.

Fernando Montaner: Pairs of quotients of Jordan pairs

4th November, 2024

In this talk, we expose ongoing joint work with I Paniello on systems of quotients (in a sense partially extending the localization theory of Jordan algebras, which in turn is inspired by the localization theory of associative algebras). Localization theory in associative algebras originated in the purpose of extending the construction of fields of quotients of integral domains, and therefore in the purpose of defining ring extensions in which a selected set of elements become invertible. As it is well known in associative theory that led to Goldie's theorems, and these in turn to more general localization theories for which the denominators of the fraction-like elements of the extensions are (one-sided) ideals taken in a class of filters (Gabriel filters). These ideas have been partially extended to Jordan algebras by several authors (starting with Zelmanov's version of Goldie theory in the Jordan setting, and its extension by Fernandez López-García Rus and Montaner) and Paniello and Montaner (among others) definition of algebras of quotients of Jordan algebras. Following the development of Jordan theory, a natural direction for extending these results is considering the context of Jordan pairs. This is the objective of the research presented here. Since obviously, a Jordan pair cannot have invertible elements unless it is an algebra, and in this case, we are back in the already developed theory, the kind of quotients that would make a significative (proper) extension of the case of algebras should be based in a different notion of the quotient. An approach that seems to be promising is considering the Jordan extension of Fountain and Gould notion of local order, as has been adapted to Jordan algebras by the work of Fernández López, and more recently by Montaner and Paniello with the notion of local order, in which the bridge between algebras and pairs is established by local algebras following the ideas of D'Amour and McCrimmon. In the talk, this idea is exposed, together with the state of the research, and the open problems that it raises.

Rongwei Yang: Linear algebra in several variables

29th October, 2024

Many mathematical and scientific problems concern systems of linear operators (A1,...,An). Spectral theory is expected to provide a mechanism for studying their properties, just like the case for an individual operator. However, defining a spectrum for non-commuting operator systems has been a difficult task. The challenge stems from an inherent problem in finite dimension: is there an analogue of eigenvalues in several variables? Or equivalently, is there a suitable notion of joint characteristic polynomial for multiple matrices A1,...,An? A positive answer to this question seems to have emerged in recent years.

Definition. Given square matrices A1,...,An of equal size, their characteristic polynomial is defined as

QA(z):=det(z0I + z1A1 + ⋯ + znAn), z=(z0,...,zn) ∈ ℂn+1.

Hence, a multivariable analogue of the set of eigenvalues is the eigensurface (or eigenvariety) Z(QA):={z ∈ ℂn+1 ∣ QA(z) = 0}. This talk will review some applications of this idea to problems involving projection matrices and finite-dimensional complex algebras. The talk is self-contained and friendly to graduate students.