# Galilean free Lie algebras

## TL;DR

This paper constructs infinite-dimensional free Lie algebras extending Galilei symmetries, which could aid in developing generalized non-relativistic gravity theories and understanding their algebraic structures.

## Contribution

It introduces a new class of infinite-dimensional free Lie algebras extending Galilei Maxwell algebras, including methods for obtaining various extensions via truncations and contractions.

## Key findings

- Constructed infinite-dimensional Galilean free Lie algebras.
- Showed how to derive extensions through truncations and contractions.
- Potential applications in non-relativistic gravity theories.

## Abstract

We construct free Lie algebras which, together with the algebra of spatial rotations, form infinite-dimensional extensions of finite-dimensional Galilei Maxwell algebras appearing as global spacetime symmetries of extended non-relativistic objects and non-relativistic gravity theories. We show how various extensions of the ordinary Galilei algebra can be obtained by truncations and contractions, in some cases via an affine Kac-Moody algebra. The infinite-dimensional Lie algebras could be useful in the construction of generalized Newton-Cartan theories gravity theories and the objects that couple to them.