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

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications Pub Date : 2025-07-08 DOI:10.1016/j.laa.2025.07.001

Francesca S. Benanti , Angela Valenti

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引用次数: 0

Abstract

Let F be a field of characteristic zero and let

V

be a variety of associative F-algebras graded by a finite abelian group G. To a variety

V

is associated a numerical sequence called the sequence of proper central G-codimensions,

c_{n}^{G, δ} (V), n \geq 1

. Here

c_{n}^{G, δ} (V)

is the dimension of the space of multilinear proper central G-polynomials in n fixed variables of any algebra A generating the variety

V

. Such sequence gives information on the growth of the proper central G-polynomials of A and in [21] it was proved that

e x p^{G, δ} (V) = \lim_{n \to \infty} \sqrt[n]{c_{n}^{G, δ} (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

e x p^{G, δ} (V) > 2

if and only if at least one of the algebras belongs to

V

Matching this result with the characterization of the varieties of almost polynomial growth given in [11], we obtain a characterization of the varieties of proper central G-exponent equal to two.

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中心指数大于2的群分级代数的变种

设F为特征为零的域，设V为由有限阿贝尔群g分级的一系列共轭F-代数，对这些代数V有一个称为中心g -协维序列的数值序列，cnG,δ(V),n≥1。这里，cnG,δ(V)是生成变量V的任意代数A的n个固定变量的多线性固有中心g多项式空间的维数。该序列给出了A的固有中心g多项式的生长信息，并在[21]中证明了expG，δ(V)=limn→∞(cnG,δ(V)n)存在，并且是一个称为固有中心g指数的整数。本文的目的是研究中心g指数大于2的共轭g级代数的变化。为此，我们构造了一个g -梯度代数的有限列表，并证明了exp，δ(V)>；2当且仅当其中至少有一个代数属于V。将这一结果与[11]中给出的几乎多项式增长的变种的刻画相匹配，我们得到了g -中心指数的变种等于2的一个刻画。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Linear Algebra and its Applications 数学-应用数学

CiteScore

2.20

自引率

9.10%

发文量

333

审稿时长

13.8 months

期刊介绍： Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.