Graded algebras with homogeneous involution and varieties of almost polynomial growth
Wesley Quaresma Cota, Felipe Yasumura
TL;DR
This work investigates the asymptotic growth of graded polynomial identities for associative algebras endowed with a homogeneous involution. By developing the framework of G-graded algebras with homogeneous involution, it defines G,*-codimensions c_n^ lat and the corresponding identities, and identifies a finite family of critical algebras whose presence governs exponential growth. The authors prove a dichotomy: any (G,*)-variety exhibits either polynomial or exponential growth, and varieties of almost polynomial growth are precisely those generated by algebras in the set I. This yields a complete classification in this setting and extends classical results on PI-algebras, graded algebras, and algebras with involution to the context of homogeneous involutions. The results have implications for understanding the structure–growth correspondence in PI-theory and for identifying the key obstructions to polynomial growth in graded involutive settings.
Abstract
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.