Quantum Real Projective Space, Disc and Sphere

Abstract

We define the $C^{\ast }$-algebra of quantum real projective space $\mathbb{R}{P}_q^2$, classify its irreducible representations and compute its $K$-theory. We also show that the $q$-disc of Klimek-Lesniewski can be obtained as a non-Galois ${\mathbb{Z}}_2$-quotient of the equator Podleś quantum sphere. On the way, we provide the Cartesian coordinates for all Podleś quantum spheres and determine an explicit form of isomorphisms between the $C^{\ast }$-algebras of the equilateral spheres and the $C^{\ast }$-algebra of the equator one.

22 pages in plain LATEX