Lie algebra representation of the 1+1-dimensional affine Poincaré group via deformation quantization

Abstract

In this work, we illustrate an explicit construction of the Lie algebra representations of the 1+1-dimensional affine Poincaré group on the cotangent bundle of cones as its coadjoint orbit. Our main construction tool is the Moyal star-product introduced to the Lie algebra of observables associated to the said group, that is, the left star-product multiplication of these observables with functions on the coadjoint orbit will give rise to a Lie algebra representations of the group. Moreover, the Lie algebra representations of the Poincaré group and the connected component affine group are recovered from the construction.