XVIII LATIN-AMERICAN ALGEBRA COLLOQUIUM MINICOURSES
Cluster algebras and cluster categories Ralf Schiffler (University of Connecticut, USA)
In this minicourse we will introduce cluster algebras, cluster categories and cluster-tilted algebras, and we will explain results that are specific to each of them as well as the connections between all three of them. The (tentative) outline of the lectures is as follows:
Lecture 1. Cluster algebras: motivation, sketch of definition, examples, cluster variables and coefficients, Laurent phenomenon, positivity conjecture, classifications.
Lecture 2.Cluster categories: definition, cluster-tilting theory, relation to cluster algebras, generalizations.
Lecture 3.Cluster-tilted algebras: various definitions, relation to tilted algebras, examples.
Lecture 4. Cluster algebras of finite mutation type: triangulated surfaces, Laurent expansion formulas, positivity theorem.
Quantum groups and Hopf algebras
Gast´on Andr´es Garc´ıa (Universidad Nacional de C´ordoba, Argentina)
Abstract. We will introduce the notion of quantum group and we will show its relation with Hopf algebras, and in particular with the classification problem of finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero.
The quantum groups, introduced in 1986 by V. Drinfeld (“Quantum groups”,Proc. Int. Congr. Math., Berkeley 1986, Vol. 1 (1987), 798-820.), form a certain class of Hopf algebras. They can be presented as deformations in one or more parameters of associative algebras related to linear algebraic groups or semisimple Lie algebras.
One of the main open problems in the theory of Hopf algebras is the classification of finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero. The first obstruction in solving the classification is the lack of enough examples. Hence, it is necessary to find new families of Hopf algebras. From the beginning, this role was played by the quantum groups. They consists of a large family with different structural properties and were used with profit to solve the classification problem for fixed dimensions.
After defining quantum groups and Hopf algebras, we will give some basic examples and we will study properties that characterize the known quantum groups. Finally, we will show how they get into the scene of the classification problem of Hopf algebras of dimensionp3and, in general, of pointed Hopf algebras with abelian coradical.
An introduction to central simple algebras and the Brauer group Eduardo Tengan (University of S˜ao Paulo, Brazil)
Central simple algebrasare finite-dimensional algebras over a fieldKwith no non-trivial two-sided ideals (they are ‘simple’) and whose center is preciselyK(they are ‘central’). Familiar examples include the ringMn(K)ofn×n matrices and division algebras (‘non-commutative fields’) overK. For a given fieldK, the central simple algebras over Kcan be organized in a group Br(K), called theBrauer groupofK, which can be thought of as a ‘directory’ of all division algebras overK. Alternatively, Br(K)can also be defined in terms of Galois cohomolgy ofK, and therefore it can be also be viewed as an arithmetic invariant ofK. The interplay between these two points of view is a very rich one, with applications not only to the theory of central simple algebras, but also to Number Theory, Field Theory, Quadratic Forms and Algebraic Geometry.
This minicourse covers some of the classical theory of central simple algebras and the Brauer group. We will start with the basic theory and then briefly review some of the results of Galois cohomology that we will need. We end up with two non-trivial computations of the Brauer group, for local and global fields.
