Explicit expressions for \vSapovalov elements in Type A

TL;DR

This paper provides explicit, determinant-based formulas for ov elements in Type A Lie algebras and superalgebras, simplifying proofs and broadening understanding of their structure and applications.

Contribution

The paper introduces new, easier proofs for explicit  expressions, extending previous results to arbitrary Borel subalgebras and interpreting them as Hessenberg matrix determinants.

Findings

Explicit formulas for  elements in Type A.

Representation-theoretic results derived from explicit formulas.

Determinant interpretation of  elements as Hessenberg matrices.

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.