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Noncommutative Finite-Dimensional Manifolds -- I. SPHERICAL MANIFOLDS AND RELATED EXAMPLES (2001)
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@MISC{Connes01noncommutativefinite-dimensional,
author = {Alain Connes and Michel Dubois-Violette},
title = { Noncommutative Finite-Dimensional Manifolds -- I. SPHERICAL MANIFOLDS AND RELATED EXAMPLES},
year = {2001}
}
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Abstract
We exhibit large classes of examples of noncommutative finitedimensional manifolds which are (non-formal) deformations of classical manifolds. The main result of this paper is a complete description of noncommutative three-dimensional spherical manifolds, a noncommutative version of the sphere S 3 defined by basic K-theoretic equations. We find a 3-parameter family of deformations of the standard 3-sphere S 3 and a corresponding 3-parameter deformation of the 4-dimensional Euclidean space R 4. For generic values of the deformation parameters we show that the obtained algebras of polynomials on the deformed R 4 u are isomorphic to the algebras introduced by Sklyanin in connection with the Yang-Baxter equation. Special values of the deformation parameters do not give rise to Sklyanin algebras and we extract a subclass, the θ-deformations, which we generalize in any dimension and various contexts, and study in some details. Here, and
Keyphrases
noncommutative finite-dimensional manifold spherical manifold related example complete description noncommutative three-dimensional spherical manifold 4-dimensional euclidean space large class obtained algebra noncommutative version corresponding 3-parameter deformation deformation parameter classical manifold main result standard 3-sphere 3-parameter family noncommutative finitedimensional manifold special value yang-baxter equation basic k-theoretic equation generic value sklyanin algebra various context
