The Heisenberg–Lorentz quantum group
Paweł KasprzakUniversity of Warsaw, Poland
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 = (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 , 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. 577–611DOI 10.4171/JNCG/67