dS–AdS Structures in Noncommutative Minkowski Spaces

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 $H_3$, $dS_3$, $AdS_3$, 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 $H_3$, $dS_3$, and $AdS_3$. We construct irreducible representations of the Minkowski spaces $H_3$ and $dS_3$. We find eigenfunctions of the Klein–Gordon equation in terms of the horospheric generators of the Minkowski spaces, and they lead to eigenfunctions on $H_3$, $dS_3$, $AdS_3$, and light cones.