Quantum Riemannian geometry of quantum projective spaces

Riemannian Geometry on Quantum Spaces

An algebraic formulation of Riemannian geometry on quantum spaces is presented, where Riemannian metric, distance, Laplacian, connection, and curvature have their counterparts. This description is also extended ∗email address: [email protected] to complex manifolds. Examples include the quantum sphere, the complex quantum projective spaces and the two-sheeted space.

Riemannian Geometry of Quantum Computation

An introduction is given to some recent developments in the differential geometry of quantum computation for which the quantum evolution is described by the special unitary unimodular group SU(2n). Using the Lie algebra su(2n), detailed derivations are given of a useful Riemannian geometry of SU(2n), including the connection, curvature, the geodesic equation for minimal complexity quantum compu...

Riemannian Geometry of Quantum Groups

Projective Quantum Spaces

Associated to the standard SUq(n) R-matrices, we introduce quantum spheres S q , projective quantum spaces CP n−1 q , and quantum Grassmann manifolds Gk(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].

Quantum Dimension and Quantum Projective Spaces

We show that the family of spectral triples for quantum projective spaces introduced by D’Andrea and Da̧browski, which have spectral dimension equal to zero, can be reconsidered as modular spectral triples by taking into account the action of the element K2ρ or its inverse. The spectral dimension computed in this sense coincides with the dimension of the classical projective spaces. The connecti...