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

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 S(3) defined by basic K-theoretic equations. We find a 3-parameter family of deformations S(u)(3) 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(u)(4) 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 S(u)(3) can span the same R(u)(4). This equivalence generates a foliation of the parameter space Sigma. This foliation admits singular leaves reduced to a point. These critical points are either isolated or fall in two I-parameter families C(+/-) subset of Sigma. 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 is an element of C(+) the above isomorphism with the Sklyanin algebra breaks down and the corresponding algebras are special cases of theta-deformations, a notion which we generalize in any dimension and various contexts, and study in some detail. 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. Besides the standard noncommutative tori, examples of theta-deformations include the recently defined noncommutative 4-sphere S(theta)(4) as well as m-dimensional generalizations, noncommutative versions of spaces R(m) and quantum groups which are deformations of various classical groups. We develop general tools such as the twisting of the Clifford algebras in order to exhibit the spherical property of the hermitian projections corresponding to the noncommutative 2n-dimensional spherical manifolds S(theta)(2n). A key result is the differential self-duality properties of these projections which generalize the self-duality of the round instanton.