“Wick Rotations”: The Noncommutative Hyperboloids and Other Surfaces of Rotations

A “Wick rotation” is applied to the noncommutative sphere to produce a noncommutative version of the hyperboloids. A harmonic basis of the associated algebra is given. It is noted that, for the one sheeted hyperboloid, the vector space for the noncommutative algebra can be completed to a Hilbert space, where multiplication is not continuous. A method of constructing noncommutative analogues of surfaces of rotation, examples of which include the paraboloid and the q-deformed sphere, is given. Also given are mappings between noncommutative surfaces, stereographic projections to the complex plane and unitary representations. A relationship with one dimensional crystals is highlighted.