# Generalization of extended Lie algebras by expansions of extended de Sitter algebra, in four-dimensions

## TL;DR

This paper explores how to generate extended Lie algebras, including Poincaré and AdS-Lorentz, through algebra expansion methods from de Sitter and conformal algebras, introducing new algebra families and their flat limits.

## Contribution

It generalizes the expansion procedure to create new extended algebra families and demonstrates their relation to known algebras via contractions.

## Key findings

- New extended algebra families $ ext{C}_k^{E}$ and $ ext{B}_k^{E}$ are constructed.
- Extended Poincaré algebra obtained as a contraction of extended de Sitter algebra.
- The expansion method applies to four-dimensional spacetime algebras.

## Abstract

Four-dimensional extended: Poincar\'e, AdS-Lorentz and Maxwell algebras, are obtained by expanding an extension of de Sitter or conformal algebra, SO(4,1) or SO(3,2). The procedure can be generalized to obtain a new family of extended $\mathcal{C}_k^{E}$ and its flat limit, the extended $\mathcal{B}_k^{E}$ algebras. The extended $\mathcal{C}_k$ and $\mathcal{B}_k$ algebras have been introduced in the literature recently.The extended Poincar\'e algebra is also obtained as an In\"on\"u-Wigner contraction of extended de Sitter algebra.