Adjoint orbits in the Lie algebra of the generalized real orthogonal group

TL;DR

This paper classifies the adjoint orbits of the generalized real orthogonal group's Lie algebra by identifying a unique representative for each orbit, enhancing understanding of its structure.

Contribution

It provides a complete classification of adjoint orbits in the Lie algebra of the generalized real orthogonal group with a unique representative for each orbit.

Findings

Explicit representatives for each orbit are identified.

The classification simplifies the understanding of the Lie algebra's structure.

The results extend classical orbit classification to a generalized setting.

Abstract

Let $(V,\gamma )$ be a real finite dimensional vector space with a symmetric bilinear form $\gamma $ whose kernel is spanned by a nonzero vector. The set of invertible real linear mappings of $(V, \gamma )$ into itself forms a Lie group called the generalized orthogonal group. This paper finds a unique representative for each orbit of the adjoint action of the generalized orthogonal group on its Lie algebra.