On the Index of Borel Subalgebras of Lie Superalgebras

Abstract

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 $\mathrm{ind}(\mathfrak{b},{\mathfrak{a}}^{\ast })=0$ 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.