Lie algebras of order F and extensions of the Poincar\'e algebra

TL;DR

This paper introduces F-Lie algebras as a generalization of Lie and Lie superalgebras, providing finite-dimensional examples, matrix realizations, and a novel extension of the Poincaré algebra through contractions of these new structures.

Contribution

It presents the first finite-dimensional examples of F-Lie algebras, matrix realizations from osp(2|m), and constructs a non-trivial Poincaré algebra extension via F-Lie algebra contractions.

Findings

Finite-dimensional F-Lie algebra examples constructed from Lie and superalgebras.

Matrix realizations of F-Lie algebras from osp(2|m).

Extension of Poincaré algebra via contraction of F-Lie algebras with F>2.

Abstract

F-Lie algebras are natural generalisations of Lie algebras (F=1) and Lie superalgebras (F=2). We give finite dimensional examples of F-Lie algebras obtained by an inductive process from Lie algebras and Lie superalgebras. Matrix realizations of the $F-$Lie algebras constructed in this way from osp(2|m) are given. We obtain a non-trivial extension of the Poincar\'e algebra by an In\"on\"u-Wigner contraction of a certain $F-$Lie algebras with $F>2$.