Hyper relative differential operators on Lie algebras

TL;DR

This paper introduces hyper relative differential operators on Lie algebras, explores their relationships with various structures, and studies hyper symplectic and hyper Hessian structures through this new lens.

Contribution

It defines hyper relative differential operators on Lie algebras and investigates their connections with DN-, KN-, and KD-structures, as well as hyper symplectic and hyper Hessian structures.

Findings

Characterization of relative differential operators using Nijenhuis operators

Relationships established between DN-, KN-, KD-structures and hyper relative differential operators

Equivalent descriptions of hyper symplectic and hyper Hessian structures

Abstract

In this paper, we first introduce the notion of a hyper relative differential operator on a Lie algebra, in which Nijenhuis operators are used to characterize the relative differential operators and their inverse. We then introduce the notions of DN-structures, KN-structures, and KD-structures on Lie algebras and study the relationships between DN-structures, KD-structures, KN-structures, and hyper relative differential operators. Finally, we investigate hyper symplectic structures and hyper Hessian structures from the view point of hyper relative differential operators, and provide equivalent descriptions for both hyper symplectic structures and hyper Hessian structures.