# The Maxwell group in 2+1 dimensions and its infinite-dimensional enhancements

## TL;DR

This paper constructs infinite-dimensional extensions of the 2+1 dimensional Maxwell group, enriching its symmetry structure with non-commutative supertranslations, and explores their geometric and physical implications.

## Contribution

It introduces a general framework for extended semi-direct products to generate infinite-dimensional Maxwell symmetries, including new geometric actions and Wess-Zumino terms.

## Key findings

- Constructed infinite-dimensional Maxwell group enhancements
- Defined coadjoint representations and geometric actions
- Identified potential applications in gravity, higher-spin theories, and quantum Hall systems

## Abstract

The Maxwell group in 2+1 dimensions is given by a particular extension of a semi-direct product. This mathematical structure provides a sound framework to study different generalizations of the Maxwell symmetry in three space-time dimensions. By giving a general definition of extended semi-direct products, we construct infinite-dimensional enhancements of the Maxwell group that enlarge the ${\rm ISL}(2,\mathbb{R})$ Kac-Moody group and the ${\rm BMS}_3$ group by including non-commutative supertranslations. The coadjoint representation in each case is defined, and the corresponding geometric actions on coadjoint orbits are presented. These actions lead to novel Wess-Zumino terms that naturally realize the aforementioned infinite-dimensional symmetries. We briefly elaborate on potential applications in the contexts of three-dimensional gravity, higher-spin symmetries, and quantum Hall systems.