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Dimas José Gonçalves

Publications and source records attributed to Dimas José Gonçalves.

5 recordsLinked to original sources
arXiv2026

Identities and isomorphisms of finite graded matrix algebras

Let $\mathbb{F}$ be a finite field and let $G$ be a group such that every finite subgroup of $G$ is cyclic. In this paper we classify the division gradings by cyclic groups on matrix algebras over $\mathbb{F}$. This yields the classification of the $G$-gradings on these algebras. Moreover if $G$ is abelian we prove that two matrix algebras over $\mathbb{F}$ with $G$-gradings satisfy the same graded identities if and only if they are isomorphic as graded algebras.

math.RA↗
arXiv2014

Minimal Varieties and Identities of Relatively Free Algebras

Let $K$ be a field of characteristic zero and let $\mathfrak{M}_5$ be the variety of associative algebras over $K$, defined by the identity $[x_1,x_2][x_3,x_4,x_5]$. It is well-known that such variety is a minimal variety and that is generated by the algebra $$A=\begin{pmatrix} E_0 & E\\ 0 & E\\ \end{pmatrix},$$ where $E=E_0\oplus E_1$ is the Grassmann algebra. In this paper, for any positive integer $k$, we describe the polynomial identities of the relatively free algebras of rank $k$ of $\mathfrak{M}_5$, \[F_k(\mathfrak{M}_5)=\dfrac{K\langle x_1,\dots, x_k \rangle}{K\langle x_1,\dots, x_k \rangle\cap T(\mathfrak{M}_5)}.\] It turns out that such algebras satisfy the same polynomial identities of some algebras used in the description of the subvarieties of $\mathfrak{M}_5$, given by Di Vincenzo, Drensky and Nardozza.

math.RA↗
arXiv2013

Multihomogeneous Normed Algebras and Polynomial Identities

In this paper we consider PI-algebras $A$ over $\R$ or $\C$. It is well known that in general such algebras are not normed algebras. In fact, there is a nilpontent commutative algebra which is not a normed algebra, see [1]. Here we address the question of whether it is possible to find a normed PI-algebra $B$ with the same polynomial identities as $A$, and moreover, whether there is some Banach PI-algebra with this property. Our main theorem provides an affirmative answer for this question and moreover we also show the existence of a Banach Algebra with the same polynomial identities as $A$. As a byproduct we prove that if $A$ is a normed PI-algebra and its completion is nil, then $A$ is nilpotent. By introducing the concept of multihomogeneous norm we obtain as an application of our main results that if $\FX$ is multihomogeneus normed algebra and $A$ is a PI-algebra such that the completion of the quotient space $\FX/Id(A)$ is nil, then $A$ is nilpotent. Both applications are extensions of the study initiated in [4].

math.RA↗
arXiv2012

Limit T-subspaces and the central polynomials in n variables of the Grassmann algebra

Let F be the free unitary associative algebra over a field F on the set X = {x_1, x_2, ...}. A vector subspace V of F is called a T-subspace (or a T-space) if V is closed under all endomorphisms of F . A T-subspace V in F is limit if every larger T-subspace W \gneqq V is finitely generated (as a T-subspace) but V itself is not. Recently Brandão Jr., Koshlukov, Krasilnikov and Silva have proved that over an infinite field F of characteristic p>2 the T-subspace C(G) of the central polynomials of the infinite dimensional Grassmann algebra G is a limit T-subspace. They conjectured that this limit T-subspace in F is unique, that is, there are no limit T-subspaces in F other than C(G). In the present article we prove that this is not the case. We construct infinitely many limit T-subspaces R_k (k \ge 1) in the algebra F over an infinite field F of characteristic p>2. For each k \ge 1, the limit T-subspace R_k arises from the central polynomials in 2k variables of the Grassmann algebra G.

math.RA↗