\v{S}apovalov elements for simple Lie algebras and basic classical simple Lie superalgebras

TL;DR

This paper investigates povalov elements for basic classical simple Lie superalgebras, establishing bounds on homomorphism dimensions between Verma modules and constructing povalov elements for isotropic roots.

Contribution

It proves a bound on the dimension of homomorphisms between Verma modules for Lie superalgebras and constructs povalov elements for isotropic roots, extending classical Lie algebra results.

Findings

povalov element construction for isotropic roots

Hom space dimension bound for Verma modules

Comparison with simple Lie algebra G

Abstract

Let $M(\gl)$ be a Verma module for a basic classical simple Lie superalgebra $\fg \neq G(3)$ defined using the distinguished Borel subalgebra, and let $\gc$ be an isotropic positive root of $\fg.$ As a special case of our first main result we show that if $\mu, \gl \in \fh^*$ with $\gl-\mu = \gc$ we have $$\dim \Hom_{\sfg}(M(\mu),M(\gl))\le 1.$$ This result applies to the construction of \v{S}apovalov elements for isotropic roots. The proof rests on a comparison with the corresponding result for a certain simple Lie algebra $G$.