Noncommutative FiniteDimensional Manifolds. I. Spherical Manifolds and Related Examples
Abstract
We exhibit large classes of examples of noncommutative finitedimensional manifolds which are (nonformal) deformations of classical manifolds. The main result of this paper is a complete description of noncommutative threedimensional spherical manifolds, a noncommutative version of the sphere S^{3} defined by basic Ktheoretic equations. We find a 3parameter family of deformations of the standard 3sphere S^{3} and a corresponding 3parameter deformation of the 4dimensional Euclidean space ^{4}. For generic values of the deformation parameters we show that the obtained algebras of polynomials on the deformed ^{4}_{u} only depend on two parameters and are isomorphic to the algebras introduced by Sklyanin in connection with the YangBaxter equation. It follows that different can span the same . 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 1parameter families . 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 the above isomorphism with the Sklyanin algebra breaks down and the corresponding algebras are special cases of θdeformations, 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 θdeformations include the recently defined noncommutative 4sphere as well as mdimensional generalizations, noncommutative versions of spaces 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 dimensional spherical manifolds . A key result is the differential selfduality properties of these projections which generalize the selfduality of the round instanton.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 2002
 DOI:
 10.1007/s0022000207152
 arXiv:
 arXiv:math/0107070
 Bibcode:
 2002CMaPh.230..539C
 Keywords:

 Mathematics  Quantum Algebra;
 General Relativity and Quantum Cosmology;
 High Energy Physics  Theory;
 Mathematical Physics
 EPrint:
 52 pages, Section 3 rewritten, references added