Journal of Noncommutative Geometry

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Volume 4, Issue 4, 2010, pp. 577–611
DOI: 10.4171/JNCG/67

Published online: 2010-09-01

The Heisenberg–Lorentz quantum group

Paweł Kasprzak[1]

(1) University 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 $\mathbb{G}$ = (A,Δ) in terms of generators α, β, γ, δAη –  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 $\mathbb G_0\subset\mathbb G$, 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.

Keywords: Quantum groups, C*-algebras

Kasprzak Paweł: The Heisenberg–Lorentz quantum group. J. Noncommut. Geom. 4 (2010), 577-611. doi: 10.4171/JNCG/67