Lie algebra representation of the two-dimensional similitude group via deformation quantization

Abstract

In this work, the construction of the Lie algebra representation of the two-dimensional similitude group will be illustrated. Instrumental to the development of this representation is the star-product on the coadjoint orbit generated by the group. It is the left star-product multiplication by Hamiltonian functions associated to the Lie algebra elements with functions on the orbit that gives rise to this representation. The Lie algebra representations of the Euclidean motion group and the connected component of the affine group will be recovered from the constructed representation. This illustrates the utility of a quantization theory to solving concrete mathematical problems.