Quantum Bundle Description of the Quantum Projective Spaces

Abstract

We realise Heckenberger and Kolb's canonical calculus on quantum projective (n-1)-space as the restriction of a distinguished quotient of the standard bicovariant calculus for Cq[SUn]. We introduce a calculus on the quantum (2n-1)-sphere in the same way. With respect to these choices of calculi, we present quantum projective (N-1)-space as the base space of two different quantum principal bundles, one with total space Cq[SUn], and the other with total space Cq[S^(2n-1)]. We go on to give Cq[CP^n] the structure of a quantum framed manifold. More specifically, we describe the module of one-forms of Heckenberger and Kolb's calculus as an associated vector bundle to the principal bundle with total space Cq[SUn]. Finally, we construct strong connections for both bundles.