Searcharxiv⌕ Search

arXiv subjects

Thiago Castilho de Mello

Publications and source records attributed to Thiago Castilho de Mello.

25 records · Page 2Linked to original sources
arXiv2016

Automorphisms of Ideals of Polynomial Rings

Let $R$ be a commutative integral domain with unit, $f$ be a nonconstant monic polynomial in $R[t]$, and $I_f \subset R[t]$ be the ideal generated by $f$. In this paper we study the group of $R$-algebra automorphisms of the $R$-algebra without unit $I_f$. We show that, if $f$ has only one root (possibly with multiplicity), then $Aut (I_f) \cong R^\times$. We also show that, under certain mild hypothesis, if $f$ has at least two different roots in the algebraic closure of the quotient field of $R$, then $Aut(I_f)$ is a cyclic group and its order can be completely determined by analyzing the roots of $f$.

math.RA↗
arXiv2015

Graded identities of block-triangular matrices

Let $F$ be an infinite field and $UT(d_1,\dots, d_n)$ be the algebra of upper block-triangular matrices over $F$. In this paper we describe a basis for the $G$-graded polynomial identities of $UT(d_1,\dots, d_n)$, with an elementary grading induced by an $n$-tuple of elements of a group $G$ such that the neutral component corresponds to the diagonal of $UT(d_1,\dots,d_n)$. In particular, we prove that the monomial identities of such algebra follow from the ones of degree up to $2n-1$. Our results generalize for infinite fields of arbitrary characteristic, previous results in the literature which were obtained for fields of characteristic zero and for particular $G$-gradings. In the characteristic zero case we also generalize results for the algebra $UT(d_1,\dots, d_n)\otimes C$ with a tensor product grading, where $C$ is a color commutative algebra generating the variety of all color commutative 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

On graded identities of block-triangular matrices with the grading of Di Vincenzo-Vasilovsky

The algebra of $n\times n$ matrices over a field $F$ has a natural $\mathbb{Z}_n$-grading. Its graded identities have been described by Vasilovsky who extended a previous work of Di Vincenzo for the algebra of $2\times 2$ matrices. In this paper we study the graded identities of block-triangular matrices with the grading inherited by the grading of $M_n(F)$. We show that its graded identities follow from the graded identities of $M_n(F)$ and from its monomial identities of degree up to $2n-2$. In the case of blocks of sizes $n-1$ and 1, we give a complete description of its monomial identities, and exhibit a minimal basis for its $T_{\mathbb{Z}_n}$-ideal.

math.RA↗
arXiv2013

On the polynomial identities of the algebra $M_{11}(E)$

Verbally prime algebras are important in PI theory. They were described by Kemer over a field $K$ of characteristic zero: 0 and $K $ (the trivial ones), $M_n(K)$, $M_n(E)$, $M_{ab}(E)$. Here $K $ is the free associative algebra of infinite rank, with free generators $T$, $E$ denotes the infinite dimensional Grassmann algebra over $K$, $M_n(K)$ and $M_n(E)$ are the $n\times n$ matrices over $K$ and over $E$, respectively. The algebras $M_{ab}(E)$ are subalgebras of $M_{a+b}(E)$, see their definition below. The generic (also called relatively free) algebras of these algebras have been studied extensively. Procesi described the generic algebra of $M_n(K)$ and lots of its properties. Models for the generic algebras of $M_n(E)$ and $M_{ab}(E)$ are also known but their structure remains quite unclear. In this paper we study the generic algebra of $M_{11}(E)$ in two generators, over a field of characteristic 0. In an earlier paper we proved that its centre is a direct sum of the field and a nilpotent ideal (of the generic algebra), and we gave a detailed description of this centre. Those results were obtained assuming the base field infinite and of characteristic different from 2. In this paper we study the polynomial identities satisfied by this generic algebra. We exhibit a basis of its polynomial identities. It turns out that this algebra is PI equivalent to a 5-dimensional algebra of certain upper triangular matrices. The identities of the latter algebra have been studied; these were described by Gordienko. As an application of our results we describe the subvarieties of the variety of unitary algebras generated by the generic algebra in two generators of $M_{11}(E)$. Also we describe the polynomial identities in two variables of the algebra $M_{11}(E)$.

math.RA↗
arXiv2012

The centre of generic algebras of small PI algebras

Verbally prime algebras are important in PI theory. They are well known over a field $K$ of characteristic zero: 0 and $K $ (the trivial ones), $M_n(K)$, $M_n(E)$, $M_{ab}(E)$. Here $K $ is the free associative algebra with free generators $T$, $E$ is the infinite dimensional Grassmann algebra over $K$, $M_n(K)$ and $M_n(E)$ are the $n\times n$ matrices over $K$ and over $E$, respectively. Moreover $M_{ab}(E)$ are certain subalgebras of $M_{a+b}(E)$, defined below. The generic algebras of these algebras have been studied extensively. Procesi gave a very tight description of the generic algebra of $M_n(K)$. The situation is rather unclear for the remaining nontrivial verbally prime algebras. In this paper we study the centre of the generic algebra of $M_{11}(E)$ in two generators. We prove that this centre is a direct sum of the field and a nilpotent ideal (of the generic algebra). We describe the centre of this algebra. As a corollary we obtain that this centre contains nonscalar elements thus we answer a question posed by Berele.

math.RA↗