The Heisenberg-Lorentz quantum group (Q1959748)

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