Asymptotics for Amitsur's Capelli-type polynomials and verbally prime PI-algebras期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Asymptotics for Amitsur's Capelli-type polynomials and verbally prime PI-algebras

Authors:	Francesca Benanti Irina Sviridova

Institution:	(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

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 $c_n \left( {M_k \left( G \right)} \right) \simeq c_n \left( {E_{k^2 ,k^2 }^ * } \right) and c_n \left( {M_{k,l} \left( G \right)} \right) \simeq c_n \left( {E_{k^2 + l^2 ,2kl}^ * } \right)$ % 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 $c_n \left( {M_k \left( F \right)} \right) \simeq c_n \left( {E_{k^2 ,0}^ * } \right)$ % 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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