*-分次Capelli多项式及其渐近性
*-Graded Capelli Polynomials and their Asymptotic
摘要：
设$F\langle Y \cup Z, \ast \rangle$是特征为零的域$F$上的自由$\ast$ -超代数，$ Γ^\ast_{M^{\pm}, L^{\pm}} $是由$\ast$ -分次Capelli多项式$Cap^{(\mathbb{Z}_2, \ast)}_{M^+} [Y^+,X]$，$Cap^{(\mathbb{Z}_2, \ast)}_{M^-} [Y^-,X]$，$Cap^{(\mathbb{Z}_2, \ast)}_{L^+} [Z^+,X]$，$Cap^{(\mathbb{Z}_2, \ast)}_{L^-} [Z^-,X]$在齐次为零的$M^+$对称变量，齐次为零的$M^-$斜变量上交替的集合生成的$T^\ast_{\mathbb{Z}_2}$ -理想，分别对齐次度为1的$L^+$对称变量和齐次度为1的$L^-$斜变量进行了研究。研究了$Γ^\ast_{M^{\pm}, L^{\pm}}.$的$\ast$ -分次余维数序列的渐近性态，特别证明了对于某些固定的自然数$M^+, M^-, L^+$和$L^-$，有限维单$\ast$ -超代数的$\ast$-分次余维数渐近等于$Γ^\ast_{M^{\pm}, L^{\pm}}$的$\ast$ -分次余维数.
英文摘要：
Let $F\langle Y \cup Z, \ast \rangle$ be the free $\ast$-superalgebra over a field $F$ of characteristic zero and let $ Γ^\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 $Γ^\ast_{M^{\pm}, L^{\pm}}.$ 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 $Γ^\ast_{M^{\pm}, L^{\pm}}$, for some fixed natural numbers $M^+, M^-, L^+$ and $L^-$.
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