q-Fuzzy spheres and quantum differentials on B_q[SU_2] and U_q(su_2)

TL;DR

This paper unifies the definitions of q-spheres and Podles spheres as q-fuzzy spheres, explores their geometric and algebraic structures, and develops covariant calculi on these quantum spaces using transmutation and twisting theory.

Contribution

It introduces a unified framework for q-spheres and Podles spheres, relating them to hyperboloids in q-Minkowski space and developing covariant calculi via transmutation and twisting.

Findings

q-spheres are defined by trace conditions similar to fuzzy spheres

Podles spheres are shown to be q-fuzzy spheres with a unified perspective

Covariant calculi on these quantum spaces are constructed using transmutation and twisting

Abstract

Whereas the classical sphere $C P^1$ can be defined as the coordinate algebra generated by the matrix entries of a projector $e$ with $\trace(e)=1$, the fuzzy-sphere is defined in the same way by $\trace(e)=1+\lambda$. We show that the standard $q$-sphere is similarly defined by $\trace_q(e)=1$ and the Podles 2-spheres by $\trace_q(e)=1+\lambda$, thereby giving a unified point of view in which the 2-parameter Podles spheres are $q$-fuzzy spheres. We show further that they arise geometrically as `constant time slices' of the unit hyperboloid in $q$-Minkowski space viewed as the braided group $B_q[SU_2]$. Their localisations are then isomorphic to quotients of $U_q(su_2)$ at fixed values of the $q$-Casimir precisely $q$-deforming the fuzzy case. We use transmutation and twisting theory to introduce a $C_q[G_C]$-covariant calculus on general $B_q[G]$ and $U_q(g)$, and use $\Omega(B_q[SU_2])$ to provide a unified point of view on the 3D calculi on fuzzy and Podles spheres. To complete the picture we show how the covariant calculus on the 3D bicrossproduct spacetime arises from $\Omega(C_q[SU_2])$ prior to twisting.