Projective quantum spaces

Abstract

Abstract. Associated to the standard SU q (n) R-matrices, we introduce quantum spheres S 2n−1 q , projective quantum spaces CP n−1 q , and quantum Grassmann manifolds G k (C n q ). These algebras are shown to be homogeneous spaces of standard quantum groups and are also quantum principle bundles in the sense of T. Brzeziński and S. Majid [1].

Geometry of the quantum complex projective spaceCP q (N)

“…A different deformed algebra of functions on the Bolyai-Lobachevskiǐ plane has been considered in [3]. The algebra of functions on complex projective space has been considered by a number of authors, see for example [4], [5] and [6]. What we have shown here is that a rich construction of differential geometry and projective geometry can be carried out on this space.…”

q -conformal invariant equations and q-plane wave solutions

“…Thus, it differs conceptually from the classical plane wave and may serve as a regularization of the latter. In the same sense it differs from the q-plane wave in the paper [11], which is not surprising, since there is used different q-Minkowski space-time (from [2,3,4] and different q-d'Alembert equation both based only on a (different) q-Lorentz algebra, and not on qconformal (or U q (sl(4))) symmetry as in our case. In fact, it is not clear whether the q-Lorentz algebra of [2,3,4] used in [11] is extendable to a q-conformal algebra.…”

Quantum Fibre Bundles. An Introduction

“…A large number of examples of quantum bundles on quantum homogeneous spaces has been found in [24]. The simplest and probably the most fundamental one is Example 2.2.4.…”

“…The other examples of quantum principal bundles constructed in [24] include: (µ, ν) may be viewed as a quotient space of SU q (2) by a coalgebra C = SU q (2)/J, where J is a right ideal in SU q (2) generated by…”