Varieties of group-graded algebras of proper central exponent greater than two

TL;DR

This work achieves a complete characterization of varieties of associative $G$-graded algebras with proper central $G$-exponent greater than two. By reducing questions to Grassmann envelopes of finite-dimensional $G\times\mathbb{Z}_2$-graded algebras and leveraging the Bahturin–Sehgal–Zaicev classification, the authors construct a finite list of graded algebras whose presence in a variety exactly detects $exp^{G,\delta}>2$. The paper then demonstrates that any such variety must contain one of these algebras, enabling a precise dichotomy: either the proper central exponent exceeds two and the variety contains one of the listed models, or it aligns with the known classification for exponent two. This yields a rigorous, lattice-structured view of the landscape of $G$-graded PI-varieties and their growth behavior, including minimal cases for exponents 3 and 4. The results integrate with prior work on almost polynomial growth and central exponents to provide a comprehensive framework for understanding the growth of proper central $G$-polynomials in graded PI-algebras.

Abstract

Let $F$ be a field of characteristic zero and let $ \mathcal V $ be a variety of associative $F$-algebras graded by a finite abelian group $G$. To a variety $ \mathcal V $ is associated a numerical sequence called the sequence of proper central $G$-codimensions, $c^{G,δ}_n(\mathcal V), \, n \ge 1.$ Here $c^{G,δ}_n(\mathcal V)$ is the dimension of the space of multilinear proper central $G$-polynomials in $n$ fixed variables of any algebra $A$ generating the variety $\mathcal V.$ Such sequence gives information on the growth of the proper central $G$-polynomials of $A$ and in \cite{LMR} it was proved that $exp^{G,δ}(\mathcal V)=\lim_{n\to\infty}\sqrt[n]{c_n^{G,δ}(\mathcal V)}$ exists and is an integer called the proper central $G$-exponent. The aim of this paper is to characterize the varieties of associative $G$-graded algebras of proper central $G$-exponent greater than two. To this end we construct a finite list of $G$-graded algebras and we prove that $exp^{G,δ}(\mathcal V) >2$ if and only if at least one of the algebras belongs to $\mathcal V$. Matching this result with the characterization of the varieties of almost polynomial growth given in \cite{GLP}, we obtain a characterization of the varieties of proper central $G$-exponent equal to two.

Theorems & Definitions (32)

• Theorem 2.1
• Theorem 2.2
• Lemma 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• Proposition 3.1
• Proposition 3.2
• proof
• Remark 3.1