Non-relativistic spin-3 symmetries in 2+1 dimensions from expanded/extended Nappi-Witten algebras

TL;DR

This paper constructs non-relativistic spin-3 symmetries in 2+1 dimensions using algebra expansions and contractions, enabling systematic development of higher-spin gravity theories in a non-relativistic setting.

Contribution

It introduces a novel method to derive non-relativistic higher-spin algebras from $ ext{sl}(3, ext{R})$ extensions via Lie algebra expansions and contractions.

Findings

Derived infinite families of non-relativistic spin-3 symmetries.

Connected higher-spin algebras to contractions of $ ext{sl}(3, ext{R})$ extensions.

Provided a systematic approach for non-relativistic higher-spin gravity theories.

Abstract

We show that infinite families of non-relativistic spin-$3$ symmetries in $2+1$ dimensions, which include higher-spin extensions of the Bargmann, Newton-Hooke, non-relativistic Maxwell, and non-relativistic AdS-Lorentz algebras, can be obtained as Lie algebra expansions of two different spin-$3$ extensions of the Nappi-Witten symmetry. These higher-spin Nappi-Witten algebras, in turn, are obtained by means of In\"on\"u-Wigner contractions applied to suitable direct product extensions of $\mathfrak{sl}(3,\mathbb{R})$. Conversely, we show that the same result can be obtained by considering contractions of expanded $\mathfrak{sl}(3,\mathbb{R})$ algebras. The method can be used to define non-relativistic higher-spin Chern-Simon gravity theories in $2+1$ dimensions in a systematic way.