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arXiv · 1510.08782

The Polynomial Part of the Codimension Growth of Affine PI Algebras

Eli Aljadeff·Geoffrey Janssens·Yakov Karasik
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Abstract

Let $F$ be a field of characteristic zero and $W$ be an associative affine $F$-algebra satisfying a polynomial identity (PI). The codimension sequence associated to $W$, $c_n(W)$, is known to be of the form $Θ(c n^t d^n)$, where $d$ is the well known (PI) exponent of $W$. In this paper we establish an algebraic interpretation of the polynomial part (the constant $t$) by means of Kemer's theory. In particular, we show that in case $W$ is a basic algebra, then $t = \frac{d-q}{2} + s$, where $q$ is the number of simple component in $W/J(W)$ and $s+1$ is the nilpotency degree of $J(W)$. Thus proving a conjecture of Giambruno.

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Eli Aljadeff, Geoffrey Janssens, Yakov Karasik. 2016-12-14. The Polynomial Part of the Codimension Growth of Affine PI Algebras. https://doi.org/10.1016/j.aim.2017.01.022

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