Program
Introductory courses (5 x 1h talks)
[1st week]
Cristina Draper -- An Exceptional Lie algebra G_2
The Killing-Cartan classification of finite-dimensional complex simple Lie algebras was one of the great milestones of 19th-century mathematics. According to it, there are four infinite families of classical simple Lie algebras (special linear, orthogonal, and symplectic) and five isolated "exceptional" examples, G_2, F_4, E_6, E_7 and E_8, of dimensions 14, 52, 78, 133 and 248 respectively. In this brief course, we would like to speak about the smallest of the exceptional algebras, G_2, as well as its relationship with another relevant nonassociative algebra, the octonion algebra, for which G_2 is the derivation algebra. We will use this example to illustrate the structure theory of simple Lie algebras over C, while giving some hints about the classification over the reals. Hopefully, we speak about the relevance of G_2 to Geometry or Physics.
Esther García González -- Relations between Lie algebras and Jordan systems [pdf, notes]
In this 5h course the students will learn different constructions connecting Lie algebras and Jordan systems, including the Tits-Kantor-Koecher construction, the Jordan algebras attached to ad-nilpotent elements of index less than or equal to three in a Lie algebra, and subquotients induced by abelian inner ideals in Lie algebra. It is intended to graduate students and researchers who are interested in learning how Jordan algebras can be used as a powerful tool to understand Lie algebras, including infinite-dimensional Lie algebras.
Reference: A. Fernández López, Jordan Structures in Lie Algebras, Mathematical Surveys and Monographs, vol.240, American Mathematical Society, Providence, RI, 2019.
Pedro Fagundes -- An introduction to zero product determined algebras [pdf, notes]
In this short course we will present the class of zero product determined algebras (zpd for short) as well as the main examples of algebras in this class. We will show that this class is stable under some classical algebraic constructions such as homomorphic images, direct sums and tensor products and also describe such algebras in the finite-dimensional associative case. If time permits, some applications of zpd algebras will also be highlighted.
Samuel Lopes -- Noncommutative algebra and representation theory: symmetry, structure & invariants [pdf, slides]
After arguing for the prominent role of symmetry in Mathematics and how Lie groups, Lie algebras and their representations naturally arise and can be understood from this perspective, the goal of this course is to introduce useful structures generalising polynomial rings and (some) enveloping algebras of Lie algebras, namely skew polynomial rings, generalized Weyl algebras and Poisson Ore extensions. We will show familiar and novel examples of these constructions and discuss several of their properties related to representations, combinatorics, growth and of course symmetry.
[2nd week]
Friedrich Wagemann -- Cohomology of Lie and Leibniz Algebras [pdf, notes] [pdf, exercices] [pdf, calculations]
We will motivate and present the basic set-up for the cohomology of Lie and Leibniz algebras. We will mention (and prove, if time permits) some classical theorems about the cohomology of semi-simple and nilpotent Lie algebras, and show how they adapt to Leibniz algebras. This last step is joint work with Jörg Feldvoss (University of South Alabama, USA).
Jorge J. Garcés -- Introduction to linear preservers.
Let A and B be two algebras. A linear preserver is a linear map T: A → B that preserves a relevant set, a relation or a quantity. Linear preservers between associative and non-associative algebras have attracted a lot of interest in the last decades. In this course we shall review some classical and modern results on linear preservers in finite dimensional C*-algebras from an elementary point of view. Along the course, we will describe (Jordan, triple) homomorphisms, isometries, zero-products and orthogonality preservers. If time allows, we will also study derivations and one-parameter groups of automorphisms.
Mykola Khrypchenko -- Incidence Algebras [pdf, notes]
Incidence algebras are a class of associative algebras which, in the finite-dimensional case, are isomorphic to certain subalgebras of the upper-triangular matrix algebra. They play an important role in combinatorics, for example, the famous Möbius function can be seen as an element of a specific incidence algebra, and the corresponding inversion formula has an interpretation in terms of the product in an incidence algebra. Our course will be dedicated to algebraic properties of incidence algebras of locally finite posets over commutative rings. We will describe the invertible elements, the idempotents, the center and the Jacobson radical of an incidence algebra. We will then show that the incidence algebra determines the poset up to an isomorphism. The automorphisms and derivations of an incidence algebra will be completely described. At the end of the course we will discuss several generalizations of incidence algebras (including non-associative ones).
Tiago Macedo -- Structure of finite-dimensional representations of map superalgebras
The purpose of this short course is to present some recent results on the representations theory of certain Lie superalgebras. The course will begin with some fundamental concepts for its sequence, such as Lie superalgebras and representations. Then, we will present the problems and results that will be the focal points of the classes. Since some techniques are needed to prove these results, we will take the opportunity to present some category theory and homological algebra techniques. Finally, we will show the main results and sketch their proofs, thus applying the techniques developed in the previous sessions.
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Research talks (1h talks)
Andrei Kukharev -- Conservative algebras of bilinear operators
Conservative algebras are a class of non-associative algebras introduced by Isaiah Kantor in 1972 during the study of a generalization of Jordan algebras. The following question naturally arises: which algebras are conservative? It is known that all associative algebras, Lie algebras, Jordan algebras, Leibniz algebras, and Zinbiel algebras are conservative. Later Kantor introduced the universal conservative algebra W(n) of bilinear operators. This algebra plays a role analogous to the role among conservative algebras that the general linear Lie algebra plays in finite-dimensional Lie algebra theory. In this talk, we will present some results on the conservative algebras W(3) and on its subalgebra W_3 of bilinear commutative operators on 3-dimensional vector space, along with the algebras of all derivations of W(3) and W_3. We have found the multiplication table and multiplication constants of the conservative algebra W(n) and W_n for arbitrary finite dimension $n$. We also found examples of commutative algebras that are not conservative, and discovered some series of conservative evolution algebras. We have established conditions under which assosymmetric algebras are conservative.
Bernardo Leite da Cunha -- Nilpotent compatible Lie algebras and their classification
In this talk we will briefly recall the notion of Lie algebra, before moving on to the concept of a compatible Lie algebra, which is a generalisation consisting of an algebra with two Lie products satisfying a certain compatibility condition. We introduce the notion of nilpotency for these structures, and we discuss a method to classify the finite-dimensional ones, for low dimensions. We will comment on the feasibility of an attempt at implementing this program using computational tools such as GAP.
Csaba Schneider -- Algebra and Geometry of Lie Algebra Invariants with Computations in Mind
Invariants of Lie algebras play an important role in many areas of mathematics, such as algebra, geometry, and even in theoretical physics. In this talk, I will outline a practical procedure that can be turned into an algorithm for the calculation of algebraically independent generators of the field of rational invariants for certain classes of Lie algebras. The procedure is based on the method of integral curves and was implemented in the system SageMath. Treating this method as a purely algebraic procedure, we obtain several interesting results on the algebra of polynomial invariants and also on the field of rational invariants. This is joint work with Vanderlei Lopes de Jesus and Igor Martins Silva.
Elkin Quintero Vanegas -- On Albert's problem and irreducible modules
Motivated by the relation between Albert’s problem and irreducible modules over the class of commutative power-associative algebras given by I. P. Shestakov, in this talk, we generalize a condition that was given for him and this generalization allows us to show a new equivalence to Albert’s problem. From this result, we can study Albert’s problem from different points of view: studying the algebras itself, or either the irreducible modules, or either the bilinear forms. Furthermore, we studied some properties of irreducible modules for the zero algebra of dimension n and we concluded that there are not irreducible modules of dimension four.
Manuela da Silva Souza -- Two-dimensional Jordan algebras: isomorphism and PI-equivalence
Let F be a field of characteristic different from 2 (finite or infinite). An algebra J over F is a Jordan algebra if ab=ba and ((aa)b)a=(aa)(ba). Jordan algebras can be associative or not. They were first introduced by Pascual Jordan (1933) to formalize the notion of an algebra of observables in quantum mechanics. Small-dimensional Jordan algebras have been extensively studied and such two-dimensional algebras have been classified (for example, over R and C). In this talk we show that any two-dimensional Jordan algebras over a finite field are isomorphic if and only if they satisfy the same polynomial identities. The same does not happen when F is infinite, even an algebraically closed field. It is a joint work with Diogo Diniz (UFCG), Dimas Gonçalves (UFSCar) and Viviane Silva (UFMG).
María Alejandra Alvarez -- On S-expansions and other transformations of Lie algebras
The aim of this work is to study the relation between S-expansions and other transformations of Lie algebras. In particular, we prove that contractions, deformations and central extensions of Lie algebras are preserved by S-expansions. We also provide several examples and give conditions so transformations of reduced subalgebras of S-expanded algebras are preserved by the S-expansion procedure. It is a joint work with J. Rosales-Gómez.
Rosa Navarro -- Solvable Leibniz Superalgebras with a given nilradical
Throughout this paper we show that under certain conditions the method for describing solvable Leibniz (resp. Lie) algebras with given nilradical by means of non-nilpotent outer derivations of the nilradical is also applicable to the case Leibniz (resp. Lie) superalgebras. Moreover, after having established the general method for Lie and Leibniz superalgebras, we classify all the solvable superalgebras on a very important class of each of them, that is, those with nilradical of maximal nilindex. Note that for (n + m)-dimensional superalgebras this maximal nilindex is n + m − 1 in the Lie case and n + m in Leibniz.