01217nas a2200097 4500008004300000245004300043210004200086520093100128100002401059856003601083 2006 en_Ud 00aSpectral geometry of k-Minkowski space0 aSpectral geometry of kMinkowski space3 a
After recalling Snyder's idea of using vector fields over a smooth manifold as "coordinates on a noncommutative space", we discuss a two dimensional toy-model whose "dual" noncommutative coordinates form a Lie algebra: this is the well known $\kappa$-Minkowski space. We show how to improve Snyder's idea using the tools of quantum groups and noncommutative geometry. We find a natural representation of the coordinate algebra of $\kappa$-Minkowski as linear operators on an Hilbert space study its "spectral properties" and discuss how to obtain a Dirac operator for this space. We describe two Dirac operators. The first is associated with a spectral triple. We prove that the cyclic integral of M. Dimitrijevic et al. can be obtained as Dixmier trace associated to this triple. The second Dirac operator is equivariant for the action of the quantum Euclidean group, but it has unbounded commutators with the algebra.
1 aD'Andrea, Francesco uhttp://hdl.handle.net/1963/2131