2025/05/12 by F. S. Benanti, Benanti, F. S., A. Valenti +1 · 1 citation
Computer Science · Mathematics · #16R50 #16W50 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Polynomial and algebraic computation #Primary 16R10 #Rings and Algebras (math.RA) #Secondary 16P90
paper · pdf · doi:10.48550/arxiv.2505.07410
openalex publication_date 2025/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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, cG,δn(\mathcal V), n ≥ 1. Here cG,δn(\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 \citeLMR it was proved that expG,δ(\mathcal V)=limn→∞√[n]cnG,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 expG,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 \citeGLP, we obtain a characterization of the varieties of proper central G-exponent equal to two.