Israel Journal of Mathematics

, Volume 156, Issue 1, pp 73–91 | Cite as

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

  • Francesca Benanti
  • Irina Sviridova


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.


Associative Algebra Young Diagram Polynomial Identity Young Tableau Jacobson Radical 
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Copyright information

© The Hebrew University Magnes Press 2006

Authors and Affiliations

  • Francesca Benanti
    • 1
  • Irina Sviridova
    • 2
  1. 1.Dipartimento di Matematica ed ApplicazioniUniversità di PalermoPalermoItaly
  2. 2.Department of Algebra and Geometric Computations Faculty of Mathematics and MechanicsUlyanovsk State UniversityUlyanovskRussia

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