Singular vector formulas for Verma modules of simple Lie superalgebras

TL;DR

This paper derives explicit formulas for singular vectors in Verma modules of simple Lie superalgebras of types BDFG, revealing new homomorphisms between these modules and advancing understanding of their structure.

Contribution

It provides explicit formulas for singular vectors in Verma modules of simple Lie superalgebras of types BDFG, establishing new homomorphisms and structural insights.

Findings

Explicit formulas for singular vectors in Verma modules

Existence of nonzero homomorphisms between Verma modules

Enhanced understanding of Lie superalgebra representations

Abstract

For a simple Lie superalgebra of type BDFG, we give explicit formulas for singular vectors in a Verma module of highest weight $\lambda - \rho$, which have weight $s_{\gamma}\lambda - \rho$ for certain positive non-isotropic roots $\gamma.$ This implies the existence of a nonzero homomorphism between the corresponding Verma modules.