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 ISL(2, ℝ) Kac-Moody group and the {hat{mathrm{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.
Highlights
(3+1)-dimensional Klein Gordon and Dirac equations associated to a particle moving in a constant electromagnetic field.1 previous indications of this symmetry were found in [5], where the kinematic symmetry of this kind of particle systems was shown to include a central extension of the translation group
The (2+1)-dimensional version of the Maxwell symmetry naturally arises when studying anyons coupled to a constant external electromagnetic field [32, 33]
We have analysed the Maxwell group in 2+1 dimensions and its infinitedimensional generalizations. This have been made by extending the notion a semi-direct product group in a way that captures the underlying structure of the Maxwell symmetry
Summary
We define a particular extension of semi-direct products, which will be used later to formulate the Maxwell group in 2+1 dimensions and its generalizations. The starting point in our construction is a semi-direct product group K ⋉σ V , where K is a Lie group, V is a vector space and σ : K → GL(V ) is a representation of K on V. The elements of a semi-direct product group are denoted by pairs (U, α), where U ∈ K and α ∈ V , and the group operation is given by [58, 59]. Some examples are the Euclidean, the Galilei and the Poincare groups [65], as well as the BMS group in three and four dimensions6 [55, 68]
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