Do Killing-Yano tensors form a Lie Algebra?

TL;DR

This paper explores whether Killing-Yano tensors form a Lie algebra under the Schouten-Nijenhuis bracket, finding that they do in constant curvature spacetimes but not generally, with implications for spacetime symmetries.

Contribution

It demonstrates the conditions under which Killing-Yano tensors form a Lie algebra and characterizes their algebraic structure in maximally symmetric spacetimes.

Findings

Killing-Yano tensors form a Lie algebra in constant curvature spacetimes.

Minkowski and (A)dS spacetimes have maximal Killing-Yano tensors.

The algebra of these tensors extends Poincare and (A)dS symmetry algebras.

Abstract

Killing-Yano tensors are natural generalizations of Killing vectors. We investigate whether Killing-Yano tensors form a graded Lie algebra with respect to the Schouten-Nijenhuis bracket. We find that this proposition does not hold in general, but that it does hold for constant curvature spacetimes. We also show that Minkowski and (anti)-deSitter spacetimes have the maximal number of Killing-Yano tensors of each rank and that the algebras of these tensors under the SN bracket are relatively simple extensions of the Poincare and (A)dS symmetry algebras.