On the orbits in the Lie algebras of some (pseudo) orthogonal groups

Abstract

A complete classification of the orbits in the Lie algebras of all the real orthogonal and pseudo-orthogonal groups of total dimension not exceeding five is presented. The classification is carried out using elementary geometrical methods, exhibiting in a clear way the relevance of the results for a lower dimensional group in obtaining the results for a higher dimensional one. For each orbit the values of the algebraic invariants are calculated, a representative element is displayed, and the geometric nature of the latter is described by listing a complete set of independent vectors invariant under it. While the orbit structure for the orthogonal groups turns out to be relatively simple, that for the Lorentz type and the de Sitter type pseudo-orthogonal groups become progressively complex. Particular care has been taken, in view of the intricacy of many of the results, to develop a suggestive and systematic notation. The orbits are classified and tabulated in a form that makes it particularly easy to apply them in practical physical problems. Examples of such problems are pointed out.