Salta al contenuto principale
Passa alla visualizzazione normale.

FRANCESCA SAVIELLA BENANTI

Asymptotics for Graded Capelli Polynomials

Abstract

The finite dimensional simple superalgebras play an important role in the theory of PI-algebras in characteristic zero. The main goal of this paper is to characterize the T 2-ideal of graded identities of any such algebra by considering the growth of the corresponding supervariety. We consider the T 2-ideal Γ M+1,L+1 generated by the graded Capelli polynomials C a p M+1[Y,X] and C a p L+1[Z,X] alternanting on M+1 even variables and L+1 odd variables, respectively. We prove that the graded codimensions of a simple finite dimensional superalgebra are asymptotically equal to the graded codimensions of the T 2-ideal Γ M+1,L+1, for some fixed natural numbers M and L. In particular csupn(Γk2+l2+1,2kl+1)≃csupn(Mk,l(F)) and csupn(Γs2+1,s2+1)≃csupn(Ms(F⊕tF)). These results extend to finite dimensional superalgebras a theorem of Giambruno and Zaicev [6] giving in the ordinary case the asymptotic equality csupn(Γk2+1,1)≃csupn(Mk(F)) between the codimensions of the Capelli polynomials and the codimensions of the matrix algebra M k (F).