*-Graded Capelli polynomials and their asymptotics

Abstract

Let F < YU Z, *> be the free *-superalgebra over a field F of characteristic zero and let Gamma(M +/-, L +/-)* be the T-Z2* - ideal generated by the set. of the s-graded Capelli polynomi- als Cap(M+)((Z2,)*())[Y+, X], Cap(L+)((Z2,)*())[Z(+), X], Cap(L-)((Z2,)*())[Z-,X] alternating on M +/- symmetric variables of homogeneous degree zero, on M- skew variables of homogeneous degree zero, on L+ symmetric variables of homogeneous degree one and on L- skew variables of homogeneous degree one, respectively. We study the asymptotic behavior of the sequence of *-graded codimensions of Gamma(M)(+/-,L)(+/-)*. In particular, we prove that the s-graded codimensions of the finite dimensional simple *-superalgebras are asymptotically equal to the *-graded codimensions of Gamma(M)(+/-,L)(+/-)*, for some fixed natural numbers M+, M-, L+ and L-.