The Polynomial Part of the Codimension Growth of Affine PI Algebras

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 $\Theta (c{n}^t{d}^n)$, 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.

24 pages

References in corpus (1)

• Kemer's Theorem for Affine PI Algebras Over a Field of Characteristic Zero