On the asymptotics for $ast$-Capelli identities

Let Fbe the free associative algebra with involution ∗ over a field of characteristic zero. If L and M are two natural numbers let Γ∗_M+1,L+1 denote theT∗-idealofFgenerated by the∗-capellipolynomialsCap+M+1,Cap−L+1 alternanting on M+1 symmetric variables and L+1skew variables,respectively.It is well known that, if F is an algebraic closed field, every finite dimensional ∗-simple algebra is isomorphic to one of the following algebras (see [4], [2]):· (Mk(F),t) with the transpose involution; · (M2m(F),s) with the symplectic involution; · (Mk(F)⊕Mk(F)op,∗) with the exchange involution. The aim of this talk is to show a relation among the asymptotics of the∗-codimensions of the finite dimensional ∗-simple algebras and the T∗-ideals Γ∗M+1,L+1, for some fixed natural numbers M and L. In particular: c∗n(Γ∗k(k+1)/2,k(k−1)/2)=c∗n((Mk(F),t)), c∗n(Γ∗m(2m−1),m(2m+1))=c∗n((M2m(F),s)) and c∗n(Γ∗k2,k2)c∗n((Mk(F)⊕Mk(F)op,∗)). Similar results have been found for simple finite dimensional superalgebras in [1] and these extend a theorem of Giambruno and Zaicev [3] giving in the ordinary case the asymptotic equality between the codimensions of the Capelli polynomials and the codimensions of the matrix algebra.This talk is based on a joint work with A. Valenti. References [1] F. Benanti, Asymptotics for Graded Capelli Polynomials, Algebra Repres. Theory18 (2015), 221–233.[2] A.GiambrunoandM.Zaicev, PolynomialIdentitiesandAsymptoticsMethods, Surveys, vol. 122, American Mathematical Society, Providence, RI, 2005. [3] A. Giambruno and M. Zaicev, Asymptotics for the Standard and the Capelli Identities, Israel J. Math.135 (2003), 125–145. [4] L.H.Rowen,PolynomialIdentitiesinRingTheory,AcademicPress,NewYork, 1980