Grau mínimo de identidades standard de álgebras de matrizes com involução graduada

Abstract

Let $F$ be a field of characteristic zero. An associative superalgebra $A = A_{0} \oplus A_{1}$ endowed with an involution $\ast$ is a $\ast$-superalgebra if $A_{0}^{\ast} = A_{0}$ and $A_{1}^{\ast} = A_{1}$. In this case, we also say that $\ast$ is a graded involution on $A$. In \cite{GSV} the authors proved that for an algebraically closed field $F$, the only graded involutions that can be defined, up to isomorphism, on the matrix superalgebras $M_{k,l}(F)$ are the transpose involution $(t)$ and the symplectic involution $(s)$, where the symplectic involution occurs only when $k = l$ and $l \neq 0$ or when $l = 0$ and $k$ is even.

In this thesis we are interested in the study of the minimality of the degree of standard identities of matrix superalgebras $M_{k,l}(F)$ endowed with graded involutions. More specifically, we provide the smallest degree of a standard $\supi$-identity in symmetric and also in skew variables of even degree of the $\ast$-superalgebra $(M_{k,l}(F),t)$ and establish upper and lower bounds for the minimal degree of a standard $\supi$-identity in symmetric and also in skew variables of odd degree of $(M_{k,l}(F),t)$. In addition, we determine the smallest degree of a standard $\supi$-identity in symmetric and also in skew variables of even degree of the $\ast$-superalgebra $(M_{k,k}(F),s)$ and we provide the smallest degree of a standard $\supi$-identity in skew variables of odd degree of $(M_{k,k}(F),s)$.