27 Brazilian Algebra Meeting · IME-USP

Speakers
Plenary Talks

Alberto Elduque

Universidad de Zaragoza, Spain

Andrea Solotar

Universidad de Buenos Aires, Argentina

Antonio Giambruno

Università degli Studi di Palermo, Italy

Cecilia Salgado

University of Groningen, Netherlands

Conchita Martínez-Pérez

University of Zaragoza, Spain

Dolors Herbera

Universitat Autònoma de Barcelona, Spain

Jeremy Rickard

University of Bristol, UK

Henrique Bursztyn

IMPA, Brazil

Volodymyr Nekrashevych

Texas A&M University, USA

Yuly Billig

Carleton University, Canada

Gradings on simple Lie algebras: old and new

After reviewing the basic definitions about gradings, it will be shown how gradings by abelian groups on a (not necessarily associative) algebra correspond to morphisms from diagonalizable group schemes to the automorphism group scheme of the algebra. This is the clue to classify gradings on simple Lie algebras. The known classification results of such gradings will be surveyed. Moreover, the notion of almost fine gradings will be introduced. The classification of gradings up to isomorphism will be shown to be equivalent to classifying almost fine gradings up to equivalence and to determining their Weyl groups. (This is based on joint work with Mikhail Kochetov.)

Alberto Elduque
Universidad de Zaragoza, Spain

Hochschild cohomology ring of triangular monomial algebras

The cup product endows the Hochschild cohomology $\mathrm{HH}^*(A)$ of an associative algebra $A$ over a field $k$ with a structure of graded commutative algebra. The description of this ring can be studied once the graded vector space $\mathrm{HH}^*(A)$ is known. There are several examples of algebras for which this ring is completely characterized. Amongst them, for triangular string algebras, quadratic string algebras, Fibonacci algebras it is known to be trivial in positive degrees. We prove that the same result holds for triangular monomial algebras. This is a result obtained in collaboration with Dalia Artenstein, Janina Letz and Amrei Oswald.

Andrea Solotar
Universidad de Buenos Aires, Argentina

Polynomial identities and central polynomials of associative algebras

In my talk I will start by giving an overview of some of the links that the theory of polynomial identities has with other branches of mathematics. Then I will focus on the progress made in the last few years in the construction of invariants of the identities of an algebra involving central polynomials. Let $A$ be an associative algebra over a field $F$ and $F\langle X\rangle$ the free associative algebra of countable rank.

Antonio Giambruno
Università degli Studi di Palermo, Italy

Mordell-Weil rank jumps on families of elliptic curves

We will give an overview of the recent developments around the variation of the Mordell-Weil rank in 1-dimensional families of elliptic curves, by studying them in the guise of elliptic surfaces. In particular, we will discuss elliptic surfaces in the light of the Enriques-Kodaira classification of algebraic surfaces, different methods to study the variation of Mordell-Weil ranks and potential obstructions due to the geometry of the underlying surface, treating rational, $K3$'s and surfaces of Kodaira dimension 1.

Cecilia Salgado
University of Groningen, Netherlands

From right angled Artin groups to Artin groups: cohomology and combinatorics

Artin groups are a fascinating and rather mysterious class of groups. Although we have a reasonably good understanding of some subfamilies, such as spherical or right angled Artin groups, there are still many basic questions about arbitrary Artin groups, such as whether they are torsion free. In this talk we will review some recent results on Artin groups, with a focus on properties of cohomological nature.

Conchita Martínez-Pérez
University of Zaragoza, Spain

Torsion free modules over commutative domains of Krull dimension $1$

Let $R$ be a commutative domain. Let $\mathcal{F}$ be the class of $R$-modules that are infinite direct sums of finitely generated torsion-free modules. In the talk we will discuss the question whether $\mathcal{F}$ is closed under direct summands.

Dolors Herbera
Universitat Autònoma de Barcelona, Spain

The finitistic dimension conjectur

The finitistic dimension conjecture, a conjecture about homological properties of representations of finite-dimensional algebras, has been open since around 1960. It has since been shown to be related to several other questions. In the last ten years or so, various new approaches to understanding the conjecture have appeared. This talk will be a survey, intended to be accessible to non-specialists, of the conjecture, its consequences, and recent developments.

Jeremy Rickard
University of Bristol, UK

Symplectic geometry on graded manifolds

Graded manifolds can be thought of as manifolds equipped with a suitable sheaf of graded algebras or, more intuitively, manifolds with local coordinates carrying an additional grading. I will discuss how symplectic geometry on graded $N$-manifolds leads to effective methods for studying objects in (higher) Lie theory and differential geometry that have gained interest in recent years, especially in connection with mathematical physics. The basic principle is that complicated/unfamiliar objects in classical geometry can often be translated into standard/familiar geometric structures defined on symplectic graded manifolds. I will illustrate fruitful applications of this viewpoint.

Henrique Bursztyn
IMPA, Brazil

Groups and algebras of dynamical origin

We will discuss algebraic objects (groups and algebras) naturally associated with topological dynamical systems. Properties of the dynamical systems can be effectively used to study algebraic properties of the associated groups and algebras such as growth, amenability, simplicity, torsion, etc. In particular, one can construct examples with exotic properties, such as groups of intermediate growth, non-elementarily amenable groups, or simple groups and algebras with additional special properties. We will also discuss how algebraic objects can be used to study properties of the associated dynamical systems.

Volodymyr Nekrashevych
Texas A&M University, USA

$AV$-modules

We wish to understand the representation theory of the Lie algebra $V$ of vector fields on a smooth algebraic variety $X$. We introduce the category of $AV$-modules, which admit compatible actions of the Lie algebra of vector fields $V$ and the commutative algebra $A$ of functions on $X$. Modules over the algebra $D$ of differential operators, or $D$-modules, form an important subcategory of $AV$-modules. Still, there are many natural examples of $AV$-modules that are not $D$-modules, like vector fields themselves. $AV$-modules were instrumental in establishing recent classification theorems of weight modules over the torus and the affine (super) spaces, however, we can study them in a much more general setting. In this talk, we shall discuss the theory of $AV$-modules over affine and projective varieties.

Yuly Billig
Carleton University, Canada