Twisted Configurations over Quantum Euclidean Spheres

Abstract

We show that the relations which define the algebras of the quantum Euclidean planes $\b{R}^N_q$ can be expressed in terms of projections provided that the unique central element, the radial distance from the origin, is fixed. The resulting reduced algebras without center are the quantum Euclidean spheres $S^{N-1}_q$. The projections $e=e^2=e^*$ are elements in $\Mat_{2^n}(S^{N-1}_q)$, with N=2n+1 or N=2n, and can be regarded as defining modules of sections of q-generalizations of monopoles, instantons or more general twisted bundles over the spheres. We also give the algebraic definition of normal and cotangent bundles over the spheres in terms of canonically defined projections in $\Mat_{N}(S^{N-1}_q)$.