Explicit expressions for \vSapovalov elements in Type A

Abstract

We give explicit expressions for \vSapovalov elements in Type A Lie algebras and superalgebras. Explicit expressions were already given in arXiv:1710.10528 Section 9, using non-commutative determinants, and in fact our first main results, Theorems 2.3 and 2.6 can be viewed as complete expansions of these determinants. But we give new proofs, which seem easier because they avoid induction and cofactor expansion. We also describe \vSapovalov elements for fgl(m,n)with respect to an arbitrary Borel subalgebra in Theorem 3.7, and interpret \vSapovalov elements in Type A as determinants of Hessenberg matrices in Theorems 4.5 and 4.9.The exact form of the explicit expressions depends on an ordering on the set of positive roots, and Hessenberg matrices are useful in changing the ordering. Having explicit expressions for \vSapovalov elements allows us to give easy proofs of several results in representation theory, see Section 6 and Subsection 4.6.