01/11 -- 13h00 - 13h50
The algebra of invariants of a complete path algebra (Auditório 208 - Pavilhão de Auditórios)
Samuel Quirino (UFMG)
In the late 70’s, Kharchenko and, independently, Lane proved that the algebra of invariants of a free algebra by the action of a homogeneous group of algebra automorphisms is a free algebra. In 2016, Cibils and Marcos extended this result for free linear k-categories and showed that the category of invariants of a free linear category of finite or tame representation type has finite or tame representation type, respectively.
In this talk we show that the algebra of invariants of a complete path algebra by the action of a homogeneous group of continuous algebra automorphisms is a complete path algebra, which inherits the representation type of the latter in case of finite or tame representation type.
01/11 -- 13h50 - 14h40
Topology of boundaries of groups (Auditório 208 - Pavilhão de Auditórios)
Lucas Henrique Rocha de Souza (UFMG)
In general, it is a difficult problem to decide if two topological spaces are or not homeomorphic. In order to try to solve this, we can use algebraic invariants such as the fundamental group. However, group theory can be quite hard as well, so we can use topological invariants to extract some algebraic information about our groups. One fruitful source of such invariants is to have a nice compactification of the group (the group seen as a discrete space) compatible with the group structure. These compactifications often come from geometric nature, the induced action of the group has a well behaved dynamic and the topology of the boundary brings information about the group, such as possible decompositions as a graph of groups, existence of free subgroups, cohomological dimension or even how our group comes from some geometric nature. In this talk, we explore some of such compactifications.
01/11 -- 14h40 - 15h30
Lie inverse semialgebras (Auditório 208 - Pavilhão de Auditórios)
José Vilca Rodriguez (USP)
It is a well-known fact that the theory of inverse semigroups is deeply related to the development of partial group actions. By considering actions by derivations of non-associative algebras and restricting them appropriately, we obtain a type of "partial action" of a Lie algebra. A naturally arising question is: Is it possible to achieve a similar approach to semigroups and partial group actions in the context of Lie algebras? This talk aims to provide a relatively satisfactory answer to this question. We will introduce the concept of Lie inverse semialgebra, which we believe captures the partial nature of these actions. Furthermore, we will show that this framework allows us to derive results analogous to those in the theory of partial group actions.
The results presented here are part of a collaborative work with M. Dokuchaev (IME-USP) and F. Johari (UFABC)