Abstract
The Poincaré algebra can be extended (non-centrally) to the Maxwell algebra and beyond. These extensions are relevant for describing particle dynamics in electromagnetic backgrounds and possibly including the backreaction due the presence of multipoles. We point out a relation of this construction to free Lie algebras that gives a unified description of all possible kinematic extensions, leading to a symmetry algebra that we call Maxwell∞. A specific dynamical system with this infinite symmetry is constructed and analysed.
Highlights
The BCR algebra has 6 generators, four translations and two of the previous Lorentz transformations
We point out a relation of this construction to free Lie algebras that gives a unified description of all possible kinematic extensions, leading to a symmetry algebra that we call Maxwell∞
The Maxwell algebra describes at same time the particle and the constant electro-magnetic background where the particle moves
Summary
We briefly review the extension of the Poincare algebra based on EilenbergChevalley cohomology. In order to determine possible extensions of this Lie algebra one can study the Chevalley-Eilenberg cohomology [21] It turns out [6] that there is a sequence of extensions of the algebra by generators that are Lorentz tensors and can be viewed as tensors of the general linear algebra gl(D) and represented by Young tableaux. We will assign level = 0 to the Lorentz generators Mab. The Poincare algebra has non-trivial cohomology that can be parametrised by the anti-symmetric tensor Zab = Z[ab] [4, 6].6. Where we have written the right-hand side in two different ways using the irreducibility constraint S[2abc,d] = 0 of the second generator arising at level = 4 This relation fixes [Zab, Zcd] = 3Sa2bd,c − Sa2bc,d − 3Sa2bc,d + Sa2bd,c = −8Sa2b[c,d]. The Lie algebra that is generated by (Mab, Pa, Zab, Yab,c, Sa1b,c,d, Sa1bc,d) will be called Maxwell. Rather than pursuing further the step by step cohomological analysis, we identify the full Lie algebraic structure in slightly different terms
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