Abstract not found

Abstract not found

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

These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments.

All authors were supported by grants from ARC (Australia) and NSERC (Canada), in addition

4. Jacobian and openness of local charts 12

(Denmark) 2 and RFBR (5-01-00629)(Russia) 4. We obtain general theorems which enable the calculation of the Dixmier trace in terms of the asymptotics of the zeta function and of the trace of the heat semigroup. We prove our results in a general semi-finite von Neumann algebra. We find for p> 1 that the asymptotics of the zeta function determines an ideal strictly larger than L p, ∞ on which the Dixmier trace may be defined. We also establish stronger versions of other results on Dixmier traces and zeta functions. 1 1.

A version of foliated spacetime is constructed in which the spatial geometry is described as a time dependent noncommutative geometry. The ADM version of the gravitational action is expressed in terms of these variables. It is shown that the vector constraint is obtained without the need for an extraneous shift vector in the action. Introduction The problem of divergences in quantum field theory, and especially in quantum gravity, strongly suggests a need to describe the geometry of space-time as something different or more general than a manifold of points. One interesting generalization of classical geometry is noncommutative differential geometry. It has attracted some attention in physics in recent years, mostly in the form of Connes-Lott and related models of particle physics [3]. It has also been used as a regularization technique for Euclidean quantum field theory [6]. Attention to applying noncommutative geometry to gravitation has been limited thus far; an expression for...

Abstract. The main result of the paper is Egorov’s theorem for transversally elliptic operators on compact foliated manifolds. This theorem is applied to describe the noncommutative geodesic flow in noncommutative geometry of Riemannian foliations.

We introduce an invariant of Riemannian geometry which measures the relative position of two von Neumann algebras in Hilbert space, and which, when combined with the spectrum of the Dirac operator, gives a complete invariant of Riemannian geometry. We show that the new invariant plays the same role with respect to the spectral invariant as the Cabibbo–Kobayashi– Maskawa mixing matrix in the Standard Model plays with respect to the list of masses of the quarks.