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Groups and noncommutative algebra: interactions and applications


The study of the interactions between group theory and non commutative algebra through group algebras is a meeting point of many important theories in algebra. Beyond group and ring theory, the subject is closely associated with representation theory of finite groups and associative algebras, and has connections with algebraic K-theory, homological algebra and algebraic topology. An the other hand, algebraic number theory plays an important role, for instance, in the study of units of integral group rings. The interactions between group theory and ring theory can be established and studied in a number of ways. Two of these are of particular interest for us. The first is as old as the abstract theory of groups itself. Given a group G and a ring of coefficients R, one can define the group ring of G over R. A. Cayley's article of 1854, which is considered to be the beginning of this abstract theory, is also the first to introduce the notion of group ring implicitly. This notion was explicitly defined by Th. Molien in 1897 and has acquired enormous importance starting with the works of E. Noether, R. Brauer and I. Schur. It was precisely through the category of modules over a group ring that the close relation between the structure theory of rings and algebras and representations of finite groups was established in these works. Recently, members of our research group have established the existence of free groups of rank 2 in the group of units of a group ring under various conditions and in the subgroup of unitary units with respect to the involution induced by the inversion map. The behavior of symmetric and skew-symmetric units under this involution has also been investigated. We intend to study this problem in its most general context, taking into account, moreover, orientation homomorphisms. The second viewpoint for the interactions between groups and rings which is of our interest consists of exhibiting groups from rings. More precisely, given a ring R, one can consider its group of units U(R) or even some of its particular subgroups. The study of the group of units of rings was traditionally concentrated on the areas of linear groups, group rings and division rings. Recently results on units of more general rings have been published. This subject if of fundamental importance both from the point of view of ring theory, where it contributes to the understanding of the structure of rings, and also from the point of view of group theory itself. There exist two famous conjectures. The first due to Lichtman which suggests that the multiplicative group of a division ring contains a free group of rank 2, and the second, due to Makar-Limanov, suggesting that a division ring infinite-dimensional and finitely generated over its center contains a free sub algebra on two generators. We intend to keep on investigating these questions and some refined versions of them, for instance, whether a non central normal subgroup of the multiplicative group of a division ring has a free subgroup of rank 2. We also intend to try and exhibit explicit constructions of free groups whenever they are known to exist. We expect to continue to look into general questions on ring theory and its connections to representation theory of algebras too. Regarding applications, there are two directions of intended work in the near future: the study of partial group actions on rings and algebraic coding theory. Partial actions and partial representation theory, besides having applications in the context of C*-algebras, where they were first introduced, have important consequences in algebra itself. Yet it has applications in the theory of JR-trees, model theory, semi group theory, topology and combinatorial theory of groups etc. We hope to continue the study of Galois theory of partial actions on commutative rings initiated recently. Projective partial representations have also been defined and it is our intention to examine their relations with projective representations of semi groups and a new type of cohomology of groups that they suggest. In regard to applications to coding theory, we would like to note that almost all of the error-correcting codes that have applications nowadays can be realized as ideals in group algebras. With respect to that, we mention a recent survey by A.V. Keralev and P. Solé published in Contemporary Mathematics. For example, it is possible to determine all codes of Reed-Solomon type over a finite field F of a given length n from the primitive idempotents of the group algebra FCn, where Cn stands for the cyclic group of order n. Presently, there has also been interest in determining categories of rings over which efficient coding theories can be constructed. We intend to determine all the primitive idempotents of certain types of finite group algebras. This can be applied to the classification of cyclic, abelian, metacyclic and nilpotent codes and to a deeper understanding of codes over other types of rings. (AU)

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Scientific publications
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
GOODAIRE, EDGAR G.; MILIES, CESAR POLCINO. Involutions and Anticommutativity in Group Rings. CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES, v. 56, n. 2, p. 344-353, . (04/15319-3, 08/57553-3)
FERRAZ, RAUL ANTONIO. Simple components of the center of FG/J(FG). COMMUNICATIONS IN ALGEBRA, v. 36, n. 9, p. 3191-3199, . (04/15319-3)

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