Noncommutative Geometry of the h-deformed Quantum Plane
Abstract
The
h
-deformed quantum plane is a counterpart of the
q
-deformed one in the set of quantum planes which are covariant under those quantum deformations of GL(2) which admit a central determinant. We have investigated the noncommutative geometry of the
h
-deformed quantum plane. There is a 2-parameter family of torsion-free linear connections, a 1-parameter sub-family of which are compatible with a skew-symmetric non-degenerate bilinear map. The skew-symmetric map resembles a symplectic 2-form and induces a metric. It is also shown that the extended
h
-deformed quantum plane is a noncommutative version of the Poincaré half-plane, a surface of constant negative Gaussian curvature.