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"Wick Rotations": The Noncommutative Hyperboloids, and other surfaces of rotations

Jonathan Gratus
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Abstract

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.

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BibTeXRIS

Jonathan Gratus. 1998-01-08. "Wick Rotations": The Noncommutative Hyperboloids, and other surfaces of rotations. https://arxiv.org/abs/math/9801036

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