JournalsjncgVol. 4, No. 4pp. 577–611

The Heisenberg–Lorentz quantum group

  • Paweł Kasprzak

    University of Warsaw, Poland
The Heisenberg–Lorentz quantum group cover
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Abstract

In this article we present a new C*-algebraic deformation of the Lorentz group. It is obtained by means of the Rieffel deformation applied to SL(2,ℂ). We give a detailed description of the resulting quantum group G\mathbb{G} = (A,Δ) in terms of generators α, β, γ, δ ∈ Aη –  the quantum counterparts of the matrix coefficients α, β, γ, δ of the fundamental representation of SL(2,ℂ). In order to construct β –   the most involved of the four generators – we first define it on the quantum Borel subgroup G0G\mathbb G_0\subset\mathbb G, then on the quantum complement of the Borel subgroup and finally we perform the gluing procedure. In order to classify representations of the C*-algebra A and to analyze the action of the comultiplication Δ on the generators α, β, γ, δ we employ the duality in the theory of locally compact quantum groups.

Cite this article

Paweł Kasprzak, The Heisenberg–Lorentz quantum group. J. Noncommut. Geom. 4 (2010), no. 4, pp. 577–611

DOI 10.4171/JNCG/67