Abdenacer Makhlouf (University of Haute Alsace, France)
Bialgebras structures, Hopf algebras and quantum groups
Bialgebras and Hopf algebras appeared first in Topology, since then applications and their popularity as a topic of study have grown tremendously. They are used in combinatorics, category theory, homological algebra, Lie groups, topology, functional analysis, quantum theory, and Hopf-Galois theory. The aim of this course is to provide an introduction to the structures of bialgebras, Hopf algebra and Quantum groups. We will discuss the motivations and provide the basic definitions and tools. Moreover, we summarize the relevant results and explore some of the applications.
Abror Khudoyberdiyev (Institute of Mathematics, Uzbekistan)
Linear operators of non-associative algebras
In this course, we discuss derivations, δ-derivations, Leibniz-derivations, quasi-derivations, generalized derivations, local and 2-local derivations, and Rota-Baxter and anti-Rota-Baxter operators of non-associative algebras. We give some common results of these operators and describe them for several important classes of non-associative algebras, such as Lie, Jordan, Leibniz, Zinbiel, and others.
Malihe Yousofzadeh (University of Isfahan, Iran)
Representation of Affine Lie Superalgebras
In this series of lectures, we will discuss the representation theory of affine Lie superalgebras. To do so, we first need to recall some general facts regarding the representation theory of finite dimensional simple Lie (super)algebras and tensor product representations. Then, we will talk about parabolically induced representations and that how we can reduce the classification of a large class of representations to the well-known classification of modules over finite dimensional simple Lie (super) algebras.
María Alejandra Alvarez (University of Antofagasta, Chile)
Free Lie algebras
The aim of this course is to provide an introduction to free Lie algebras. We will review the basics on free Lie algebras starting with enveloping algebras and the Poincaré-Birkhoff-Witt Theorem. We will study algebraic properties and some of the constructions of bases for free Lie algebras.
Samuel Lopes (University of Porto, Portugal)
Automorphisms, Derivations and Cohomology of classical and Quantum Algebras
Polynomial algebras, differential operator algebras and enveloping algebras of Lie algebras, along with their quantum versions, are ubiquitous in almost all branches of mathematics as well as quantum physics. Their structure, representation theory, combinatorics and geometric features are of pivotal importance and have shaped fundamental trends in modern research, feeding on outstanding open problems, such as the Jacobian and Dixmier conjectures. The purpose of this course is to study these objects from the point of view of their symmetries, surveying recent methods which have been proven successful in the study of their automorphisms, derivations and Hochschild cohomology.
Stéphane Launois (University of Caen, France)
Quantum algebras and beyond
The aim of this course is to provide an introduction to quantum algebras (mainly quantized coordinate rings) and their semiclassical limits. We will introduce recent algorithmic methods to study the representation theory of these algebras and will explain their link with various combinatorial objects (including totally nonnegative matrices).
Tiago Macedo (Federal University of São Paulo, Brazil)
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.
Yunhe Sheng (Jilin University, China)
Deformations and Cohomologies of Algebraic Structures
In this course first, we introduce deformations and cohomologies of Lie algebras with applications. Then we apply the same idea to other types of algebraic structures, such as Rota-Baxter Lie algebras, difference Lie algebras, and embedding tensors. If time permits, we will also talk about the global objects corresponding to these structures.