paper

Noncommutative Manifolds the Instanton Algebra and Isospectral Deformations

arXiv:math/0011194 · doi:10.1007/PL00005571

Abstract

We give new examples of noncommutative manifolds that are less standard than the NC-torus or Moyal deformations of $\Rb^n$. They arise naturally from basic considerations of noncommutative differential topology and have non-trivial global features. The new examples include the instanton algebra and the NC-4-spheres . The noncommutative algebras $\Ac=C^{\ify} (S^{4}_θ)$ of functions on NC-spheres are solutions to the vanishing, , of the Chern character in the cyclic homology of $\Ac$ of an idempotent $e \in M_4 (\Ac), e^2 = e, e = e^*$. The universal noncommutative space defined by this equation is a noncommutative Grassmanian defined by very non trivial cubic relations. This space contains the suspension of a NC-3-sphere intimately related to quantum group deformations of but for unusual values (complex values of modulus one) of the parameter of -analogues, $q=\exp (2πi \t)$. We then construct the noncommutative geometry of $S_{\t}^4$ as given by a spectral triple $(\Ac, \Hc, D)$ and check all axioms of noncommutative manifolds. The Dirac operator on the noncommutative 4-spheres $S_{\t}^4$ gives a solution to the basic quartic equation defining the `volume form' $ < (e - {1/2}) [D,e]^4 > = \g_5$, where is the projection on the commutant of $4 \ts 4$ matrices. Finally, we show that any compact Riemannian spin manifold whose isometry group has rank admits isospectral deformations to noncommutative geometries.

We introduce the notion of admissible morphism and use it to clarify a confusing point. Definite version to appear in CMP. 20 pages, latex

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