Asymptotics for Graded Capelli Polynomials

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Abstract

The finite dimensional simple superalgebras play an important role in the theoryof PI-algebras in characteristic zero. The main goal of this paper is to characterize the T2-ideal of graded identities of any such algebra by considering the growth of the correspondingsupervariety. We consider the T2-ideal M+1,L+1 generated by the graded Capelli polynomialsCapM+1[Y,X] and CapL+1[Z,X] alternanting on M + 1 even variables and L + 1odd variables, respectively.We prove that the graded codimensions of a simple finite dimensionalsuperalgebra are asymptotically equal to the graded codimensions of the T2-ideal M+1,L+1, for some fixed natural numbers M and L. In particularcsupn ( k2+l2+1,2kl+1) csupn (Mk,l(F ))andcsupn ( s2+1,s2+1) csupn (Ms(F ⊕ tF)).These results extend to finite dimensional superalgebras a theorem of Giambruno andZaicev [6] giving in the ordinary case the asymptotic equalitycsupn ( k2+1,1) csupn (Mk(F ))between the codimensions of the Capelli polynomials and the codimensions of the matrixalgebra Mk(F ).
Lingua originaleEnglish
pagine (da-a)221-233
Numero di pagine13
RivistaAlgebras and Representation Theory
Volume18
Stato di pubblicazionePublished - 2015

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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