Upper block triangular matrix algebras graded by finite cyclic groups: the factorability of their graded T-ideals and the minimal varieties

• Autor(a) principal: Marcos Antônio da Silva Pinto
• Data de Publicação: 2020
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Repositório Institucional da UFMG
• Texto Completo: http://hdl.handle.net/1843/38503
• topic: Graded algebras; Finite cyclic groups; Factorability; Minimal varieties; Matemática – Teses.; Algebra abstrata – Teses.; Grupos finitos – Teses.; Fatoração (Matematica) – Teses.; Variedades (Matematica) – Teses.

Let $F$ be an algebraically closed field of characteristic zero and $G$ be a finite cyclic group. In this work, all the $F$-algebras are assumed to be associative. Given finite dimensional $G$-simple $F$-algebras $A_{1},\ldots,A_{m}$, taken as graded subalgebras of matrix algebras with some elementary gradings, consider the upper block triangular matrix algebra $A:=(UT(A_{1},\ldots,A_{m}), \widetilde{\alpha})$ endowed with an elementary $G$-grading induced by a map $\widetilde{\alpha}$ (defined by gluing the gradings of the $A_{i}$'s). In this thesis, we approach two main topics: the \emph{factoring property} related to the $T_{G}$-ideal $\mathrm{Id}_{G}(A)$ of the $G$-graded polynomial identities satisfied by $A$ and the \emph{minimal varieties} of associative $G$-graded PI-algebras over $F$, of finite basic rank, with respect to a given $G$-exponent. More precisely, we prove that any finite dimensional $G$-simple $F$-algebra, previously described by Bahturin, Sehgal and Zaicev (for any arbitrary group), can be seen, for cyclic groups, as a graded subalgebra of a matrix algebra endowed with an elementary grading. Moreover, if $G$ is a cyclic $p$-group, with $p$ being an arbitrary prime, we establish that $\mathrm{Id}_{G}(A)$ is factorable if, and only if, there exists at most one index $i\in\{1,\ldots,m\}$ such that $A_{i}$ is not $G$-regular if, and only if, there exists a unique isomorphism class of $G$-gradings for $A$. This is a generalization of the results presented by Avelar, Di Vincenzo and da Silva, when $G$ has order 2, which already contrasted with the ordinary case, investigated by Giambruno and Zaicev. It is worth highlighting that we use different techniques from those employed in such cases. Still, by generalizing the concept of $G$-regularity, we introduce the definition of \emph{$\alpha$-regularity} and we establish nice connections between such concept and the so-called \emph{invariance subgroups}. Finally, when $G$ is not necessarily a $p$-group, we present necessary and sufficient conditions in order to obtain that $\mathrm{Id}_{G}((UT(A_{1},A_{2}),\widetilde{\alpha}))$ is factorable, by requiring that $A_{1}$ and $A_{2}$ are $\alpha_{1}$-regular and $\alpha_{2}$-regular, respectively. Regarding the minimal varieties, we prove that they are generated by suitable $G$-graded upper block triangular matrix algebras $(UT(A_{1},\ldots,A_{m}), \widetilde{\alpha})$. On the other hand, by assuming some conditions over these algebras, we show that the varieties generated by some of them are minimal. These problems was explored, in ordinary case, by Giambruno and Zaicev, and, when $G$ is of prime order, by Di Vincenzo, da Silva and Spinelli.