AI 中文总结
本文针对复数域上基础经典李超代数的Borel子代数，通过扩展强正交根与Kostant cascade理论到广义根系理论，给出其指标的上界并证明相关指标为0的结论。
AI 中文摘要
设$\boldsymbol{\frak{b}=\frak{h}\boldsymbol{\frak{n}}$是复数域$\boldsymbol{\frak{C}}$上基础经典李超代数的Borel子代数，其中$\boldsymbol{\frak{h}}$为Cartan子代数。本文给出$\boldsymbol{\frak{b}}$的指标$\boldsymbol{\text{ind}(\boldsymbol{\frak{b}})}$的上界，部分情形下该上界为0，此时$\boldsymbol{\text{ind}(\boldsymbol{\frak{b}})=0}$。此外，证明$\boldsymbol{\text{ind}(\boldsymbol{\frak{b}},\boldsymbol{\frak{n}})=0}$，由此推出对$\boldsymbol{\frak{b}}$的所有理想$\boldsymbol{\frak{i} \boldsymbol{\frak{n}}}$，均有$\boldsymbol{\text{ind}(\boldsymbol{\frak{b}},\boldsymbol{\frak{i}})=0}$；还证明对$\boldsymbol{\frak{b}}$的所有交换理想$\boldsymbol{\frak{a} \boldsymbol{\frak{n}}}$，均有$\boldsymbol{\text{ind}(\boldsymbol{\frak{b}},\boldsymbol{\frak{a}}^*)=0}$。这些结果通过将强正交根理论与Kostant cascade理论扩展到Dimitrov和Fioresi提出的广义根系理论而得到。
英文摘要
Let $\mathfrak{b}=\mathfrak{h}\oplus\mathfrak{n}$ be a Borel subalgebra of a basic classical Lie superalgebra over $\mathbb{C}$ with $\mathfrak{h}$ a Cartan subalgebra. We give an upper bound for the index $\mathrm{ind}(\mathfrak{b})$ of $\mathfrak{b}$; in some instances this bound is $0$, in which case $\mathrm{ind}(\mathfrak{b})=0$. Additionally we prove that $\mathrm{ind}(\mathfrak{b},\mathfrak{n})=0$, which implies $\mathrm{ind}(\mathfrak{b},\mathfrak{i})=0$ for all ideals $\mathfrak{i} \subseteq \mathfrak{n}$ of $\mathfrak{b}$. We also show that $\mathrm{ind}(\mathfrak{b},\mathfrak{a}^*)=0$ for all abelian ideals $\mathfrak{a} \subseteq \mathfrak{n}$ of $\mathfrak{b}$. These results are achieved by extending the theory of strongly orthogonal roots and the Kostant cascade to the theory of generalized root systems developed by Dimitrov and Fioresi.
