Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples

Original · EN

We exhibit large classes of examples of noncommutative finite-dimensional 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 S³ defined by basic K-theoretic equations. We find a 3-parameter family of deformations S³ᵤ of the standard 3-sphere S³ and a corresponding 3-parameter deformation of the 4-dimensional Euclidean space R⁴. For generic values of the deformation parameters we show that the obtained algebras of polynomials on the deformed R⁴ᵤ only depend on two parameters and are isomorphic to the algebras introduced by Sklyanin in connection with the Yang-Baxter equation. It follows that different S³ᵤ can span the same R⁴ᵤ. This equivalence generates a foliation of the parameter space Σ. This foliation admits singular leaves reduced to a point. These critical points are either isolated or fall in two 1-parameter families C±⊂ Σ. Up to the simple operation of taking the fixed algebra by an involution, these two families are identical and we concentrate here on C₊. For u ∈ C₊ the above isomorphism with the Sklyanin algebra breaks down and the corresponding algebras are special cases of θ-deformations, a notion which we generalize in any dimension and various contexts, and study in some details. Here, and this point is crucial, the dimension is not an artifact, i.e. the dimension of the classical model, but is the Hochschild dimension of the corresponding algebra which remains constant during the deformation....