Finite-dimensional Lie algebras of order F

TL;DR

This paper introduces a method to construct finite-dimensional F-Lie algebras for F>2, expanding the known examples and providing matrix representations, with applications to extensions of the Poincaré algebra.

Contribution

It presents an inductive construction of finite-dimensional F-Lie algebras for F>2, including explicit matrix realizations and applications to algebra extensions.

Findings

Constructed finite-dimensional F-Lie algebras for F>2

Provided matrix realizations from classical Lie algebras and superalgebras

Extended the Poincaré algebra via contractions of F-Lie algebras

Abstract

$F-$Lie algebras are natural generalisations of Lie algebras (F=1) and Lie superalgebras (F=2). When $F>2$ not many finite-dimensional examples are known. In this paper we construct finite-dimensional $F-$Lie algebras $F>2$ by an inductive process starting from Lie algebras and Lie superalgebras. Matrix realisations of $F-$Lie algebras constructed in this way from $\mathfrak{su}(n), \mathfrak{sp}(2n)$ $\mathfrak{so}(n)$ and $\mathfrak{sl}(n|m)$, $\mathfrak{osp}(2|m)$ are given. We obtain non-trivial extensions of the Poincar\'e algebra by In\"on\"u-Wigner contraction of certain $F-$Lie algebras with $F>2$.