arXiv subjects
Publications and source records attributed to Wesley Quaresma Cota.
We classify, up to PI-equivalence, associative algebras over a field of characteristic zero whose codimension sequences have quadratic growth. More precisely, making use of fundamental algebras, we prove that every such algebra is PI-equivalent to a finite direct sum of algebras generating minimal varieties of at most quadratic codimension growth, together with a possible nilpotent summand.
Let $A$ be an associative algebra with a superinvolution $*$ over a field of characteristic zero, and let $c_n^*(A)$, $n = 1, 2, \ldots$, denote its sequence of $*$-codimensions. It is well known that this sequence is either polynomially bounded or grows exponentially. In the polynomial case, a central problem in PI-theory is the classification of varieties ${V}$ for which $c_n^*({V}) \approx αn^k$ for a given $k$. One of the main objectives of this paper is to classify minimal varieties of unitary algebras endowed with a superinvolution that exhibit quadratic codimension growth. We obtain a structural characterization, up to PI-equivalence, of all unitary algebras with quadratic codimension growth. As a consequence, we show that any unitary variety of quadratic codimension growth is generated by a direct sum of algebras generating minimal varieties.
Let $A$ be an associative algebra graded by a finite group $G$ over a field ${F}$ of characteristic zero. One associates to $A$ the sequence of $G$-graded codimensions $c_n^G(A)$, $n=1,2,\ldots$, which measures the growth of the polynomial identities satisfied by $A$. It is known that this sequence is either polynomially bounded or grows exponentially. In this paper, we study unitary $G$-graded varieties of polynomial codimension growth. In particular, we classify the varieties generated by unitary algebras with quadratic codimension growth and show that these varieties can be described as a direct sums of algebras that generate minimal $G$-graded varieties.
The theory of algebras with polynomial identities has developed significantly, with special attention devoted to the classification of varieties according to the asymptotic behavior of their codimension sequences. This sequence is a fundamental numerical invariant, as it captures the growth rate of the polynomial identities of a given algebra. Special partial classification results have been obtained, with particular interest devoted to algebras equipped with additional structure. In this paper, we consider associative G-graded algebras endowed with a graded involution. We provide a complete classification, up to equivalence, of unitary algebras with quadratic codimension growth. Our approach establishes a direct correspondence between the algebras generating minimal varieties and the nonzero multiplicities appearing in the decomposition of the proper cocharacters. As a consequence, we establish that every variety with at most quadratic growth is generated by an algebra that decomposes as a direct sum of algebras generating minimal varieties of at most quadratic growth.
For an associative algebra $A$ over a field of characteristic zero, let $P_n(A)$ and $P_n^z(A)$ denote the spaces of multilinear polynomials of degree $n$ modulo the polynomial identities and the central polynomials of $A$, respectively. We also write $Δ_n(A)$ for the space of multilinear central polynomials of degree $n$ modulo the polynomial identities of $A$. The corresponding sequences of colengths, central colengths and proper central colengths measure the number of irreducible components in the $S_n$-module decompositions of $P_n(A)$, $P_n^z(A)$ and $Δ_n(A)$, respectively. In this paper, we investigate several examples of PI-algebras and explicitly describe their cocharacter, central cocharacter and proper central cocharacter sequences. As a consequence, we obtain a complete classification, up to PI-equivalence, of all algebras whose sequences of colengths and central colengths are bounded by a constant.
Subalgebras of upper triangular matrix algebras have played a fundamental role in the classification of minimal varieties of polynomial growth. Such classification has become a source of study in recent years since it leads to the more general classification of varieties of polynomial growth $n^k$, as has already been proven in many contexts for several values of $k$. In this paper, we study the asymptotic behavior of the sequence of codimensions of algebras graded by a finite group $G$ and endowed with a graded involution $*$, also called $(G,*)$-algebras. We classify the minimal varieties generated by a finite-dimensional $(G,*)$-algebra with quadratic growth.
In recent years, many results have been established regarding classifications of varieties whose colength sequences are bounded by a fixed constant. In this work, we explore this theme in the setting of algebras endowed with a graded involution, called $(G,*)$-algebras. We give an explicit description of the decomposition of the $\langle n \rangle$-cocharacter for some important $(G,*)$-algebras $A$, for every $\langle n \rangle=(n_1, \ldots, n_{2t})$. For each algebra $A$, the $n$th colength is defined as the number of irreducible components that appear in these decompositions. Our aim is to classify varieties whose $n$th colengths are bounded by a fixed constant.
An important aspect in the theory of algebras with polynomial identities is the study of the asymptotic behavior of the codimension sequence $c_n(A),\, n\geq 1,$ which measures the growth of polynomial identities of a given algebra $A$. In this context, graded identities naturally arise as prominent tools, since ordinary polynomial identities can be viewed as a particular case of graded identities. Moreover, as an involution does not necessarily preserve the homogeneous components of a grading, it is natural to consider the notion of a homogeneous involution. In this work, we investigate the behavior of the codimension sequence in the setting of $G$-graded algebras endowed with a homogeneous involution. More specifically, we characterize the varieties of polynomial growth in terms of the exclusion of a list of algebras from the variety. As a consequence, we provide the classification of the varieties with almost polynomial growth in this setting.