The polynomial part of the codimension growth of affine PI algebras

Aljadeff, E.;Janssens, Geoffrey;Karasik, Y.
(2017) Advances in mathematics — Vol. 309, n° 1, p. 487-511 (2017)

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Let F be a field of characteristic zero and W an associative affine F-algebra satisfying a polynomial identity (PI). The codimension sequence {cn(W)} associated to W is known to be of the form Θ(ntdn), 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 (hence finite dimensional), t=q−d2+s, where q is the number of simple component in W/J(W) and s+1 is the nilpotency degree of J(W) (the Jacobson radical of W). Thus proving a conjecture of Giambruno. © 2017 Elsevier Inc.
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Aljadeff, E., Janssens, G., & Karasik, Y. (2017). The polynomial part of the codimension growth of affine PI algebras. Advances in mathematics, 309(1), 487-511. https://doi.org/10.1016/j.aim.2017.01.022 (Original work published 2017)