dS-AdS Structures in Noncommutative Minkowski Spaces

Abstract

We consider a family of noncommutative four-dimensional Minkowski spaces with the signature (1, 3) and two types of spaces with the signature (2, 2). The Minkowski spaces are defined by the common reflection equation and differ in anti-involutions. There exist two Casimir elements, and. xing one of them leads to the noncommutative “ homogeneous” spaces H₃, dS₃, AdS₃, and light cones. We present a semiclassical description of the Minkowski spaces. There are three compatible Poisson structures: quadratic, linear, and canonical. Quantizing the first leads to the Minkowski spaces. We introduce horospheric generators of the Minkowski spaces, and they lead to the horospheric description of H₃, dS₃, and AdS₃. We construct irreducible representations of the Minkowski spaces H₃ and dS₃. We find eigenfunctions of the Klein-Gordon equation in terms of the horospheric generators of the Minkowski spaces, and they lead to eigenfunctions on H₃, dS₃, AdS₃, and light cones.