A New One-Parameter Invariant Function for Algebras

Abstract

Invariant functions are very useful tools for the study of contractions of algebras. Hrivnák and Novotný (J Geom Phys 58(2):208–217, 2008) introduced the $\psi$ and $\varphi$ one-parameter invariant functions and by taking their procedure into consideration we introduced in 2016 the invariant two-parameter function $\bar{\psi }.$ In this communication we introduce a new one-parameter invariant function for algebras, the $\upsilon$ function, which is related with $\bar{\psi }.$ We compute the values of this new function for several types of algebras, particularly filiform Lie algebras and Malcev algebras, and for the algebra induced by the Lorentz group SO(3),  which allows us to prove that the n-dimensional classical-mechanical model built upon certain types of n-dimensional Lie algebras cannot be obtained as a limit process of a quantum-mechanical model based on a n-dimensional Heisenberg algebra, for certain values of n. By using the symbolic computation package SAGE as a tool, we also conjecture that $\upsilon \ge \psi$ for any algebra, which is indeed true for algebras of lower dimensions.