*-Graded Capelli Polynomials and their Asymptotic

TL;DR

This paper investigates the asymptotic growth of $ extit{*-graded}$ codimensions in free $ extit{*-superalgebras}$, establishing their equivalence with those of certain finite-dimensional simple $ extit{*-superalgebras}$, and introduces $ extit{*-graded}$ Capelli polynomials.

Contribution

It introduces and analyzes $ extit{*-graded}$ Capelli polynomials and proves the asymptotic equivalence of codimensions between free $ extit{*-superalgebras}$ and finite-dimensional simple $ extit{*-superalgebras}$.

Findings

Asymptotic equality of $ extit{*-graded}$ codimensions for free and finite-dimensional simple $ extit{*-superalgebras}$

Development of $ extit{*-graded}$ Capelli polynomials with specific symmetry and skew-symmetry properties

Characterization of the asymptotic behavior of $ extit{*-graded}$ codimensions

Abstract

Let $F\langle Y \cup Z, \ast \rangle$ be the free $\ast$-superalgebra over a field $F$ of characteristic zero and let $ \Gamma^\ast_{M^{\pm}, L^{\pm}} $ be the $T^\ast_{\mathbb{Z}_2}$-ideal generated by the set of the $\ast$-graded Capelli polynomials $Cap^{(\mathbb{Z}_2, \ast)}_{M^+} [Y^+,X]$, $Cap^{(\mathbb{Z}_2, \ast)}_{M^-} [Y^-,X]$, $Cap^{(\mathbb{Z}_2, \ast)}_{L^+} [Z^+,X]$, $Cap^{(\mathbb{Z}_2, \ast)}_{L^-} [Z^-,X]$ 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 $\Gamma^\ast_{M^{\pm}, L^{\pm}}.$ 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 $\Gamma^\ast_{M^{\pm}, L^{\pm}}$, for some fixed natural numbers $M^+, M^-, L^+$ and $L^-$.