Asymptotics for Amitsur's Capelli-type polynomials and verbally prime PI-algebras |
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Authors: | Francesca Benanti Irina Sviridova |
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Affiliation: | (1) Dipartimento di Matematica ed Applicazioni, Università di Palermo, via Archirafi, 34, 90123 Palermo, Italy;(2) Department of Algebra and Geometric Computations Faculty of Mathematics and Mechanics, Ulyanovsk State University, 432970 Ulyanovsk, Russia |
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Abstract: | We consider associativePI-algebras over a field of characteristic zero. The main goal of the paper is to prove that the codimensions of a verbally prime algebra [11] are asymptotically equal to the codimensions of theT-ideal generated by some Amitsur's Capelli-type polynomialsE M,L * [1]. We recall that two sequencesa n,b nare asymptotically equal, and we writea n ≃b n,if and only if lim n→∞(a n/b n)=1.In this paper we prove that % MathType!End!2!1!, whereG is the Grassmann algebra. These results extend to all verbally primePI-algebras a theorem of A. Giambruno and M. Zaicev [9] giving the asymptotic equality % MathType!End!2!1! between the codimensions of the matrix algebraM k(F) and the Capelli polynomials. The second author is partially supported by grants RFFI 04-01-00739a, E02-2.0-26. |
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