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SPACETIME FROM SYMMETRY: THE MOYAL PLANE FROM THE POINCARÉ–HOPF ALGEBRA

A. P. BALACHANDRAN

Department of Physics, Syracuse University, Syracuse, NY 13244-1130, USA

Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911 Leganés, Madrid, Spain

Cátedra de Excelencia

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and

M. MARTONE

Department of Physics, Syracuse University, Syracuse, NY 13244-1130, USA

Dipartimento di Scienze Fisiche, University of Napoli and INFN, Via Cinthia I-80126 Napoli, Italy

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https://doi.org/10.1142/S0217732309031144Cited by:4 (Source: Crossref)

Abstract

We show how to get a noncommutative product for functions on spacetime starting from the deformation of the coproduct of the Poincaré group using the Drinfel'd twist. Thus it is easy to see that the commutative algebra of functions on spacetime (ℝ⁴) can be identified as the set of functions on the Poincaré group invariant under the right action of the Lorentz group provided we use the standard coproduct for the Poincaré group. We obtain our results for the noncommutative Moyal plane by generalizing this result to the case of the twisted coproduct. This extension is not trivial and involves cohomological features.

As is known, spacetime algebra fixes the coproduct on the diffeomorphism group of the manifold. We now see that the influence is reciprocal: they are strongly tied.

Keywords:

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