arXiv subjects
Publications and source records attributed to Luiz Henrique de Souza Matos.
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.
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.