Conference abstracts

Session S05 - Rings and Algebras

No date set

## Anticommutativity of Symmetric and Skew-symmetric Elements under Generalized Oriented Involutions

### UFBA - Universidade Federal da Bahia, Brazil   -   edward.landi@ufba.br

Given an involution $*$ in a group ring $RG$, we can define the sets $(RG)_*=\left\{\alpha\in RG:\alpha^*=\alpha\right\}$ and $(RG)_*^-=\left\{\alpha\in RG:\alpha^*=-\alpha\right\}$, called the set of symmetric and skew-symmetric elements, respectively. Under certain conditions in $R$, $G$, or the involution in $RG$, many authors proved that some identities satisfied in these sets could be lifted to the entire group ring, and, in some cases, given the impossibility of such lifting, they describe the basic structures of the group ring $RG$.

Generalizing the results found in [GP13a, GP13b, GP14], using a group homomorphism $\sigma:G\rightarrow \mathcal{U}(R)$, we will define and explore the involution $\sigma*:RG\rightarrow RG$, called generalized oriented involution, exposing the group structures, as well as the ring conditions, such that $(RG)_{\sigma*}$ or $(RG)_{\sigma*}^-$ be anticommutative.

\begin{thebibliography}{5} \bibitem[BP06]{BP06}{BROCHE CRISTO, O.; POLCINO MILIES, C. Symmetric elements under orientated involutions in group rings, \begin{itshape}Communications in Algebra\end{itshape}, v. 34, n. 9, p. 3347-3356, 2006.}

\bibitem[GP13a]{GP13a}{GOODAIRE, E. G.; POLCINO MILIES, C. Involutions and Anticommutativity in Group Rings. Em: {\it Canadian Mathematical Bulletin}, v. 52, n. 2, p. 344-353, 2013.}

\bibitem[GP13b]{GP13b}{GOODAIRE, E. G.; POLCINO MILIES, C. Oriented Involutions and Skew-symmetric Elements in Group Rings. Em: {\it Journal of Algebra and Its Applications}, v. 12, n. 1, 2013.}

\bibitem[GP14]{GP14}{GOODAIRE, E. G.; POLCINO MILIES, C. Oriented Group Involutions and Anticommutativity in Group Rings. Em: {\it Communications in Algebra}, v. 42, n. 4, p. 1657-1667, 2014.}

\bibitem[JM06]{JM06}{JESPERS, E.; RUIZ MARÍN, M. On symmetric elements and symmetric units in group rings, {\it Communications in Algebra}, v. 34, n. 2, p. 727-736, 2006.}

\bibitem[PT15a]{PT15a}{PETIT LOBAO, T. C.; TONUCCI, E. L. Anticommutativity of Skew-symmetric Elements under Generalized Oriented Involutions, preprint (2015) arXiv:1511.06907v1 [math.RA].}

\bibitem[PT15b]{PT15b}{PETIT LOBAO, T. C.; TONUCCI, E. L. Anticommutativity of Symmetric Elements under Generalized Oriented Involutions, preprint (2015) arXiv:1510.06004v1 [math.RA].}

\bibitem[V13]{V13}{VILLA, A. H. {\it Involuções de grupo orientadas em álgebras de grupo}. 2013. 76f. Ph.D thesis - Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, 2013.} \end{thebibliography}

Joint work with Thierry Petit Lobao (Universidade Federal da Bahia, Brazil).