Quantum Real Projective Space, Disc and Sphere

TL;DR

This paper introduces the $C^*$-algebra of quantum real projective space, classifies its irreducible representations, computes its $K$-theory, and relates the $q$-disc to the Podleś quantum sphere, expanding the understanding of quantum geometric structures.

Contribution

It defines the $C^*$-algebra of quantum real projective space, classifies its irreducible representations, and establishes a connection between the $q$-disc and Podleś quantum sphere.

Findings

Classified irreducible representations of $ ext{C}^*( ext{RP}_q^2)$

Computed the $K$-theory of quantum real projective space

Related the $q$-disc to the Podleś quantum sphere as a non-Galois $ ext{Z}_2$-quotient

Abstract

We define the $C^*$-algebra of quantum real projective space $\R P_q^2$, 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 $\Z_2$-quotient of the equator Podle\'s quantum sphere. On the way, we provide the Cartesian coordinates for all Podle\'s quantum spheres and determine an explicit form of isomorphisms between the $C^*$-algebras of the equilateral spheres and the $C^*$-algebra of the equator one.