The Heisenberg–Lorentz quantum group

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 $\mathbb{G}$ = (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 ${\mathbb{G}}_0\subset \mathbb{G}$, 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.