We consider associative PI-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 the T-ideal generated by some Amitsur’s Capelli-type polynomials E¤M;L [1]. In particular we prove that cn(Mk(G)) '' cn(E¤k2;k2 ) and cn(Mk;l(G)) '' cn(E¤k2+l2;2kl); where G is the Grassmann algebra. These results extend to all verbally prime PI-algebras a theorem of A.Giambruno and M.Zaicev [9] giving the asymptotic equality cn(Mk(F)) '' cn(E¤k2;0) between the codimensions of the matrix algebra Mk(F) and the Capelli polynomials.
BENANTI, F., SVIRIDOVA, I. (2006). Asymptotics for the Amitsur's Capelli - Type Polynomials and Verbally Prime PI-Algebras. ISRAEL JOURNAL OF MATHEMATICS, 156(1), 73-91 [10.1007/BF02773825].
Asymptotics for the Amitsur's Capelli - Type Polynomials and Verbally Prime PI-Algebras
BENANTI, Francesca Saviella;SVIRIDOVA, Irina
2006-01-01
Abstract
We consider associative PI-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 the T-ideal generated by some Amitsur’s Capelli-type polynomials E¤M;L [1]. In particular we prove that cn(Mk(G)) '' cn(E¤k2;k2 ) and cn(Mk;l(G)) '' cn(E¤k2+l2;2kl); where G is the Grassmann algebra. These results extend to all verbally prime PI-algebras a theorem of A.Giambruno and M.Zaicev [9] giving the asymptotic equality cn(Mk(F)) '' cn(E¤k2;0) between the codimensions of the matrix algebra Mk(F) and the Capelli polynomials.File | Dimensione | Formato | |
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