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

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.