Central polynomials of associative algebras and their growth

Giambruno, Antonio; Zaicev, Mikhail

doi:10.1090/proc/14172

Central polynomials of associative algebras and their growth
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by Antonio Giambruno and Mikhail Zaicev PDF

Proc. Amer. Math. Soc. 147 (2019), 909-919 Request permission

Abstract:

A central polynomial for an algebra $A$ is a polynomials in noncommutative variables taking central values in $A$. If an algebra has central polynomials, e.g., the algebra of $k\times k$ matrices, can one measure how many are there?

Here we study the growth of central polynomials for any algebra satisfying a polynomial identity over a field of characteristic zero. We prove the existence of two limits called the central exponent and the proper central exponent of $A$. They give a measure of the exponential growth of the central polynomials and the proper central polynomials of $A$. They are comparable with $exp(A)$, the PI-exponent of the algebra.

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