In 1984, Regev started the quantitative study of the space of central polynomials by computing the exponential rate of growth of central polynomials of matrix algebras. More generally, for <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3689_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, one considers the dimension <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3689_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$c_n^{\delta }(A)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>c</mi> <mi>n</mi> <mi>δ</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of the space of multilinear central polynomials of degree n modulo the polynomial identities of an algebra A. In 2018, Giambruno and Zaicev proved the limit <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="209_2025_3689_Article_IEq3.gif" Format="GIF" Height="28" Rendition="HTML" Resolution="72" Type="Linedraw" Width="93" /> </InlineMediaObject> <EquationSource Format="TEX">$\lim \limits _{n \rightarrow \infty }\root n \of {c_n^{\delta }(A)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <munder> <mo movablelimits="false">lim</mo> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </munder> <mroot> <mrow> <msubsup> <mi>c</mi> <mi>n</mi> <mi>δ</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> <mi>n</mi> </mroot> </mrow> </math></EquationSource> </InlineEquation> exists and it is an integer. In this paper we consider such a situation for superalgebras endowed with a superinvolution or a graded involution and present the existence of the corresponding limit.

Proper central exponent of superalgebras with graded involution or superinvolution

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发表日期：

2025年2月12日

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D. La Mattina,
R. B. dos Santos,
A. C. Vieira

摘要

In 1984, Regev started the quantitative study of the space of central polynomials by computing the exponential rate of growth of central polynomials of matrix algebras. More generally, for $n \ge 1$ $n \geq 1$ , one considers the dimension $c_n^{\delta }(A)$ $c_{n}^{δ} (A)$ of the space of multilinear central polynomials of degree n modulo the polynomial identities of an algebra A. In 2018, Giambruno and Zaicev proved the limit $\lim \limits _{n \rightarrow \infty }\root n \of {c_n^{\delta }(A)}$ $lim_{n \to \infty} \sqrt[n]{c_{n}^{δ} (A)}$ exists and it is an integer. In this paper we consider such a situation for superalgebras endowed with a superinvolution or a graded involution and present the existence of the corresponding limit.