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.

22 pages, Latex2e, Section 3.1 revised, References changed

References in corpus (1)

• Quantum Lobachevsky Planes

Cited by in corpus (14)

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• Quantum Mechanics on the h-deformed Quantum Plane
• Representation Functions for Jordanian Quantum Group SL_h(2) and Jacobi Polynomials
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• Classification of bicovariant differential calculi on the Jordanian quantum groups GL_{g,h}(2) and SL_{h}(2) and quantum Lie algebras
• Automorphisms of associative algebras and noncommutative geometry
• The Gervais-Neveu-Felder equation for the Jordanian quasi-Hopf U_{h;y}(sl(2)) algebra
• The coloured quantum plane
• Gauge Theories on Open Lie Algebra Non-Commutative Spaces
• Properties of $2\times 2$ h-deformed quantum (super)matrices
• Two-Parameter Differential Calculus on the h-Exterior Plane