Irreducible Killing and conformal Killing tensors on homogeneous plane waves

TL;DR

The paper advances the classification of hidden symmetries on four-dimensional homogeneous plane waves by computing irreducible 2-tensors that are Killing or conformal Killing, using the first BGG machinery on homogeneous spaces. It develops explicit prolongation data and executes a detailed parameter-case analysis (generic vs exceptional) to enumerate irreducible tensors, including nine- and six-dimensional families in specific metrics. The results distinguish conformally flat from non-flat cases, provide explicit tensor formulas (via Propositions) and connect CKTs to KTs when possible, thereby enriching understanding of Lorentzian symmetry structures and their geometric implications. The work has implications for Lorentzian holonomy, exact solutions in General Relativity, and the broader study of hidden symmetries in curved spacetimes, with precise algebraic and geometric descriptions enabled by computer algebra. Finally, it clarifies which homogeneous plane waves carry nontrivial irreducible conformal Killing or Killing tensors and how they relate under conformal and isometric equivalences.

Abstract

This paper presents a classification of irreducible Killing and conformal Killing 2-tensors on homogeneous plane waves, a specific class of Lorentzian metrics on four-dimensional manifolds. Using the framework of BGG operators, we derive explicit formulae for these tensors and identify the conditions under which they exist.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Proposition 2.1
• proof
• Proposition 3.1
• proof