Non integral exponential growth of central polynomials

Abstract

Let A be an algebra over a field F of characteristic zero. For every $n\ge 1$, let $\delta _n(A)$ be the number of linearly independent multilinear proper central polynomials of A in n fixed variables. It was shown in [8] that if A is a finite dimensional associative algebra, the limit $\delta (A)=\lim _{n\rightarrow \infty }\root n \of {\delta _n(A)}$ always exists and is an integer. Here we show that such a result cannot be extended in general to non associative algebras. In fact we construct a five-dimensional non associative algebra such that the above limit exists and $\delta (A)\approx 3.61$, a non integer.