On the Leibniz bracket, the Schouten bracket and the Laplacian

Details On the Leibniz bracket, the Schouten bracket and the Laplacian On the Leibniz bracket, the Schouten bracket and the Laplacian

2003-06-22

arxiv.org/abs/gr-qc/0306102v1

J.Math.Phys.45:2405-2410,2004

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

Scientific paper

10.1063/1.1738188

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.

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