The Heisenberg-Lorentz quantum group

TL;DR

This paper introduces a new quantum deformation of the Lorentz group using Rieffel deformation on SL(2,C), providing a detailed algebraic and representation-theoretic analysis of the resulting quantum group.

Contribution

It presents the first C*-algebraic deformation of the Lorentz group via Rieffel deformation, with explicit generator descriptions and representation classification.

Findings

Constructed the Heisenberg-Lorentz quantum group explicitly.

Classified representations of the quantum group's C*-algebra.

Analyzed the action of comultiplication on generators.

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 in terms of generators - the quantum counterparts of the matrix coefficients of the fundamental representation of SL(2,C). In order to construct the most involved of 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 of the Heisenberg-Lorentz quantum group and to analyze the action of the comultiplication on the generators we employ the duality in the theory of locally compact quantum groups.