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Claudemir Fidelis

Publications and source records attributed to Claudemir Fidelis.

6 recordsLinked to original sources
arXiv2021

$\mathbb{Z}$-graded identities of the Lie algebras $U_1$ in characteristic 2

Let $K$ be any field of characteristic two and let $U_1$ and $W_1$ be the Lie algebras of the derivations of the algebra of Laurent polynomials $K[t,t^{-1}]$ and of the polynomial ring $K[t]$, respectively. The algebras $U_1$ and $W_1$ are equipped with natural $\mathbb{Z}$-gradings. In this paper, we provide bases for the graded identities of $U_1$ and $W_1$, and we prove that they do not admit any finite basis.

math.RA↗
arXiv2021

A note on $\mathbb{Z}$-gradings on the Grassmann algebra and Elementary Number Theory

Let $E$ be the Grassmann algebra of an infinite dimensional vector space $L$ over a field of characteristic zero. In this paper, we study the $\mathbb{Z}$-gradings on $E$ having the form $E=E_{(r_{1},r_{2}, r_{3})}^{(v_{1},v_{2}, v_{3})}$, in which each element of a basis of $L$ has $\mathbb{Z}$-degree $r_{1}$, $r_{2}$, or $r_{3}$. We provide a criterion for the support of this structure to coincide with a subgroup of the group $\mathbb{Z}$, and we describe the graded identities for the corresponding gradings. We strongly use Elementary Number Theory as a tool, providing an interesting connection between this classical part of Mathematics, and PI Theory. Our results are generalizations of the approach presented in [11].

math.RA↗
arXiv2021

$\mathbb{Z}$-graded identities of the Lie algebras $U_1$

Let $K$ be an infinite field of characteristic different from two and let $U_1$ be the Lie algebra of the derivations of the algebra of Laurent polynomials $K[t,t^{-1}]$. The algebra $U_1$ admits a natural $\mathbb{Z}$-grading. We provide a basis for the graded identities of $U_1$ and prove that they do not admit any finite basis. Moreover, we provide a basis for the identities of certain graded Lie algebras with a grading such that every homogeneous component has dimension $\leq 1$, if a basis of the multilinear graded identities is known. As a consequence of this latter result we are able to provide a basis of the graded identities of the Lie algebra $W_1$ of the derivations of the polynomial ring $K[t]$. The $\mathbb{Z}$-graded identities for $W_1$, in characteristic 0, were described in \cite{FKK}. As a consequence of our results, we give an alternative proof of the main result, Theorem 1, in \cite{FKK}, and generalize it to positive characteristic. We also describe a basis of the graded identities for the special linear Lie algebra $sl_q(K)$ with the Pauli gradings where $q$ is a prime number.

math.RA↗
arXiv2020

$\mathbb{Z}$-gradings of full support on the Grassmann algebra

Let $E$ be the infinite dimensional Grassmann algebra over a field $F$ of characteristic zero. In this paper we investigate the structures of $\mathbb{Z}$-gradings on $E$ of full support. Using methods of elementary number theory, we describe the $\mathbb{Z}$-graded polynomial identities for the so-called $2$-induced $\mathbb{Z}$-gradings on $E$ of full support. As a consequence of this fact we provide examples of $\mathbb{Z}$-gradings on $E$ which are PI-equivalent but not $\mathbb{Z}$-isomorphic. This is the first example of graded algebras with infinite support that are PI-equivalent and not isomorphic as graded algebras. We also present the notion of central $\mathbb{Z}$-gradings on $E$ and we show that its $\mathbb{Z}$-graded polynomial identities are closely related to the $\mathbb{Z}_{2}$-graded polynomial identities of $\mathbb{Z}_{2}$-gradings on $E$.

math.RA↗
arXiv2017

Graded Isomorphisms on Upper Block Triangular Matrix Algebras

We describe the graded isomorphisms of rings of endomorphisms of graded flags over graded division algebras. As a consequence describe the isomorphism classes of upper block triangular matrix algebras (over an algebraically closed field of characteristic zero) graded by a finite abelian group.

math.RA↗