*-Graded Capelli Polynomials and their Asymptotic

Abstract

Let $F⟨Y\cup Z,\ast ⟩$ be the free $\ast$-superalgebra over a field $F$ of characteristic zero and let ${\Gamma }_{M^{\pm },L^{\pm }}^{\ast }$ be the $T_{{\mathbb{Z}}_2}^{\ast }$-ideal generated by the set of the $\ast$-graded Capelli polynomials $Ca{p}_{M^{+}}^{({\mathbb{Z}}_2,\ast )}[Y^{+},X]$, $Ca{p}_{M^{-}}^{({\mathbb{Z}}_2,\ast )}[Y^{-},X]$, $Ca{p}_{L^{+}}^{({\mathbb{Z}}_2,\ast )}[Z^{+},X]$, $Ca{p}_{L^{-}}^{({\mathbb{Z}}_2,\ast )}[Z^{-},X]$ 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 $\ast$-graded codimensions of ${\Gamma }_{M^{\pm },L^{\pm }}^{\ast }.$ In particular we prove that the $\ast$-graded codimensions of the finite dimensional simple $\ast$-superalgebras are asymptotically equal to the $\ast$-graded codimensions of ${\Gamma }_{M^{\pm },L^{\pm }}^{\ast }$, for some fixed natural numbers $M^{+},M^{-},L^{+}$ and $L^{-}$.