On the Leibniz bracket, the Schouten bracket and the Laplacian

TL;DR

This paper explores the properties of the Leibniz bracket in graded algebras, establishing its relation to Lie algebra structures and generalizing the Laplacian for exterior forms.

Contribution

It introduces a fundamental theorem linking Leibniz brackets of operators to their commutators and extends the Laplacian expression to arbitrary exterior forms.

Findings

Leibniz bracket relates to the commutator of operators.

Under certain conditions, it induces a graded Lie algebra structure.

Generalization of the Laplacian for exterior forms.

Abstract

The Leibniz bracket of an operator on a (graded) algebra is defined and some of its properties are studied. A basic theorem relating the Leibniz bracket of the commutator of two operators to the Leibniz bracket of them, is obtained. Under some natural conditions, the Leibniz bracket gives rise to a (graded) Lie algebra structure. In particular, those algebras generated by the Leibniz bracket of the divergence and the Laplacian operators on the exterior algebra are considered, and the expression of the Laplacian for the product of two functions is generalized for arbitrary exterior forms.