Proper central exponent of superalgebras with graded involution or superinvolution

Abstract

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 >= 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 1$$\end{document}, one considers the dimension cn delta(A)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_n<^>{\delta }(A)$$\end{document} 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 limn ->infinity cn delta(A)n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lim \limits _{n \rightarrow \infty }\root n \of {c_n<^>{\delta }(A)}$$\end{document} 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.