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).