The Invariant Two-Parameter Function of Algebras ψ

Abstract

At present, the research on invariant functions for algebras is very extended since Hrivnak and Novotný defined in 2007 the invariant functions ψ and φ as a tool to study the Inonu–Wigner contractions (IW-contractions), previously introduced by those authors in 1953. In this paper, we introduce a new invariant two-parameter function of algebras, which we call ψ ¯ , as a tool which makes easier the computations and allows researchers to deal with contractions of algebras. Our study of this new function is mainly focused in Malcev algebras of the type Lie, although it can also be used with any other types of algebras. The main goal of the paper is to prove, by means of this function, that the five-dimensional classical-mechanical model built upon certain types of five-dimensional Lie algebras cannot be obtained as a limit process of a quantum-mechanical model based on a fifth Heisenberg algebra. As an example of other applications of the new function obtained, its computation in the case of the Lie algebra induced by the Lorentz group S O ( 3 , 1 ) is shown and some open physical problems related to contractions are also formulated.