Coefficients of \v{S}apovalov elements for simple Lie algebras and contragredient Lie superalgebras

Abstract

We provide upper bounds on the degrees of the coefficients of \v{S}apovalov elements for a simple Lie algebra. If $\fg$ is a contragredient Lie superalgebra and $\gc$ is a positive isotropic root of $\fg,$ we prove the existence and uniqueness of the \v{S}apovalov element for $\gc$ and we obtain upper bounds on the degrees of their coefficients. For type A Lie superalgebras we give a closed formula for \v{S}apovalov elements. Often the coefficients of \v{S}apovalov elements are products of linear factors, and we provide some reasons for this coming from representation theory. We also explore the relationships between \v{S}apovalov elements coming from different roots, and their behavior when the Borel subalgebra is changed.