The Heisenberg-Lorentz quantum group

TY - JOUR

T1 - The Heisenberg-Lorentz quantum group

AU - Kasprzak, Pawel Lukasz

PY - 2010

Y1 - 2010

N2 - 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 G= (A,Δ) in terms of generators α̂β̂γ ̂ δ̂ ∈ Aη the quantum counterparts of the matrix coefficients α,βγ of the fundamental representation of SL.(2, ℂ). In order to construct β̂ - the most involved of the four generators - we first define it on the quantum Borel subgroup G0 ⊂ 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 α̂β̂γ̂, δ̂ we employ the duality in the theory of locally compact quantum groups.

AB - 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 G= (A,Δ) in terms of generators α̂β̂γ ̂ δ̂ ∈ Aη the quantum counterparts of the matrix coefficients α,βγ of the fundamental representation of SL.(2, ℂ). In order to construct β̂ - the most involved of the four generators - we first define it on the quantum Borel subgroup G0 ⊂ 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 α̂β̂γ̂, δ̂ we employ the duality in the theory of locally compact quantum groups.

M3 - Journal article

SN - 1661-6952

VL - 4

SP - 577

EP - 611

JO - Journal of Noncommutative Geometry

JF - Journal of Noncommutative Geometry

ER -