The Heisenberg–Lorentz quantum group

Abstract

In this article we present a new C${}^{\ast }$-algebraic deformation of the Lorentz group. It is obtained by means of the Rieffel deformation applied to $\mathrm{SL}(2,C)$. We give a detailed description of the resulting quantum group $\mathbb{G}=(A,\Delta )$ in terms of generators $\hat{\alpha },\hat{\beta },\hat{\gamma },\hat{\delta }\in A^{\eta }$ –  the quantum counterparts of the matrix coefficients $\alpha ,\beta ,\gamma ,\delta$ of the fundamental representation of $\mathrm{SL}(2,C)$. In order to construct $\hat{\beta }$ –   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${}^{\ast }$-algebra $A$ and to analyze the action of the comultiplication $\Delta$ on the generators $\hat{\alpha },\hat{\beta },\hat{\gamma },\hat{\delta }$ we employ the duality in the theory of locally compact quantum groups.