Non integral exponential growth of central polynomials
A \textit{central polynomial} for an (associative) algebra \(A\) is a polynomial in noncommutative variables taking central values in \(A\). \textit{A. Regev} started the quantitative study of the space of central polynomials by computing the exponential rate of growth of central polynomials for \(k \times k\) matrices [Commun. Algebra 44, No. 10, 4411--4421 (2016; Zbl 1354.16023)]. Let \(n\ge 1\), one considers the space \(P_n\) of multilinear polynomials in \(n\) fixed variables modulo the central polynomials of \(A\) and calls \(c^z_n (A)\) the corresponding dimension. Let \(c_n (A)\) be the dimension of \(P_n\) modulo the polynomial identities of \(A\) and \(\delta_n (A)\) the dimension of the central polynomials in \(P_n\) modulo the polynomial identities of \(A\). One has \(c_n(A) = c^z_n(A)+\delta_n (A)\) for all \(n\ge 1\). Regev's result was extended as follows [the first and third authors, Isr. J. Math. 226, No. 1, 15--28 (2018; Zbl 1432.16024)]. Given a finite dimensional associative algebra \(A\) over an algebraically closed field of characteristic zero, either \(\delta_n (A) = 0\) for all \(n \ge C\), for some constant \(C\), or \(\delta_n (A)\) grows polynomially or \(\delta_n (A)\) grows exponentially and the following limit exists and is a positive integer: \[ \delta(A)=\lim_{n\to\infty}\sqrt[n]{\delta_n(A)}. \] Now, the authors show that the result above cannot be extended in general to non-associative algebras. They construct a five-dimensional non-associative algebra such that the above limit exists and \(\delta(A)\approx 3.61\), a non integer.
scientific article; zbMATH DE number 7021300