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Luís Felipe Gonçalves Fonseca

Publications and source records attributed to Luís Felipe Gonçalves Fonseca.

9 recordsLinked to original sources
arXiv2020

$\mathbb{Z}_3$-graded identities of the pair $(M_3(K),gl_3(K))$

Let $M_n(K)$ be the algebra of $n \times n$ matrix over an infinite integral domain $K$. Let $gl_n(K)$ be the Lie algebra of $n \times n$ matrix with the usual Lie product over $K$. Let $G = \{g_1,\ldots,g_n\}$ be a group of order $n$. We describe the polynomials that form a basis for the $G$-graded identities of the pair $(M_n(K),gl_n(K))$ with an elementary $G$-grading induced by the $n$-tuple $(g_1,\ldots,g_n)$. In the end, we describe an explicit basis for the $\mathbb{Z}_3$-graded identities of the pair $(M_3(K),gl_3(K))$.

math.RA↗
arXiv2020

A note on the $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded identities of $E \otimes E$ over a finite field

Let $F$ be a finite field of $char F = p$ and size $|F| = q$. Let $E$ be the unitary infinity dimensional Grassmann algebra. In this short note, we describe the $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded identities of $E_{k^{*}}\otimes E$, where $E_{k^{*}}$ is the Grassmann algebra with a specific $\mathbb{Z}_{2}$-grading. In the end, we discuss about the $\mathbb{Z}_2 \times \mathbb{Z}_2$-graded GK-dimension of $E_{k^*}\otimes E$ in $m$ variables.

math.RA↗
arXiv2020

On graded polynomial identities of $sl_{2}(F)$ over a finite field

Let $F$ be a finite field of $char F > 3$ and $sl_{2}(F)$ be the Lie algebra of traceless $2\times 2$ matrices over $F$. This paper aims for the following goals: Find a basis for the $\mathbb{Z}_{2}$-graded identities of $sl_{2}(F)$; Find a basis for the $\mathbb{Z}_{3}$-graded identities of $sl_{2}(F)$ when $F$ contains a primitive 3rd root of one; Find a basis for the $\mathbb{Z}_{2}\times \mathbb{Z}_{2}$-graded identities of $sl_{2}(F)$.

math.RA↗
arXiv2020

$\mathbb{Z}_{2}$-graded identities of the Grassmann algebra over a finite field

Let $F$ be a finite field with characteristic $p > 2$ and let $G$ be the unitary Grassmann algebra generated by an infinite dimensional vector space $V$ over $F$. In this paper, we determine a basis of the $\mathbb{Z}_{2}$-graded polynomial identities for any non-trivial $\mathbb{Z}_{2}$-grading such that a basis of $V$ is homogeneous in this grading.

math.RA↗
arXiv2017

Graded Polynomial Identities for Matrices with the Transpose Involution over an Infinite Field

Let $F$ be an infinite field, and let $M_{n}(F)$ be the algebra of $n\times n$ matrices over $F$. Suppose that this algebra is equipped with an elementary grading whose neutral component coincides with the main diagonal. In this paper, we find a basis for the graded polynomial identities of $M_{n}(F)$ with the transpose involution. Our results generalize for infinite fields of arbitrary characteristic previous results in the literature which were obtained for the field of complex numbers and for a particular class of elementary G-gradings.

math.RA↗
arXiv2017

Degree-inverting involutions on matrix algebras

Let $F$ be an algebraically closed field of characteristic zero, and $G$ be a finite abelian group. If $A=\oplus_{g\in G} A_g$ is a $G$-graded algebra, we study degree-inverting involutions on $A$, i.e., involutions $*$ on $A$ satisfying $(A_g)^*\subseteq A_{g^{-1}}$, for all $g\in G$. We describe such involutions for the full $n\times n$ matrix algebra over $F$ and for the algebra of $n\times n$ upper triangular matrices.

math.RA↗
arXiv2014

Graded polynomial identities and central polynomials of matrices over an infinite integral domain

Let $K$ be an infinite integral domain and $M_{n}(K)$ be the algebra of all $n\times n$ matrices over $K$. This paper aims for the following goals: Find a basis for the graded identities for elementary grading in $M_{n}(K)$ when the neutral component and diagonal coincide; Describe the $\mathbb{Z}_{p}$-graded central polynomials of $M_{p}(K)$ when $p$ is a prime number; Describe the $\mathbb{Z}$-graded central polynomials of $M_{n}(K)$.

math.RA↗