5 research outputs found
The Spectral Geometry of the Equatorial Podles Sphere
We propose a slight modification of the properties of a spectral geometry a
la Connes, which allows for some of the algebraic relations to be satisfied
only modulo compact operators. On the equatorial Podles sphere we construct
suq2-equivariant Dirac operator and real structure which satisfy these modified
properties.Comment: 6 pages. Latex. V2: Minor changes; to appear in Comptes Rendus
Mathematiqu
Metrics and Pairs of Left and Right Connections on Bimodules
Properties of metrics and pairs consisting of left and right connections are
studied on the bimodules of differential 1-forms. Those bimodules are obtained
from the derivation based calculus of an algebra of matrix valued functions,
and an SL\sb q(2,\IC)-covariant calculus of the quantum plane plane at a
generic and the cubic root of unity. It is shown that, in the
aforementioned examples, giving up the middle-linearity of metrics
significantly enlarges the space of metrics. A~metric compatibility condition
for the pairs of left and right connections is defined. Also, a compatibility
condition between a left and right connection is discussed. Consequences
entailed by reducing to the centre of a bimodule the domain of those conditions
are investigated in detail. Alternative ways of relating left and right
connections are considered.Comment: 16 pages, LaTeX, nofigure
The Isospectral Dirac Operator on the 4-dimensional Orthogonal Quantum Sphere
Equivariance under the action of Uq(so(5)) is used to compute the left
regular and (chiral) spinorial representations of the algebra of the orthogonal
quantum 4-sphere S^4_q. These representations are the constituents of a
spectral triple on this sphere with a Dirac operator which is isospectral to
the canonical one on the round undeformed four-sphere and which gives metric
dimension four for the noncommutative geometry. Non-triviality of the geometry
is proved by pairing the associated Fredholm module with an `instanton'
projection. We also introduce a real structure which satisfies all required
properties modulo smoothing operators.Comment: 40 pages, no figures, Latex. v2: Title changed. Sect. 9 on real
structure completely rewritten and results strengthened. Additional minor
changes throughout the pape
Local Index Formula on the Equatorial Podles Sphere
We discuss spectral properties of the equatorial Podles sphere. As a
preparation we also study the `degenerate' (i.e. ) case (related to the
quantum disk). We consider two different spectral triples: one related to the
Fock representation of the Toeplitz algebra and the isopectral one. After the
identification of the smooth pre--algebra we compute the dimension
spectrum and residues. We check the nontriviality of the (noncommutative) Chern
character of the associated Fredholm modules by computing the pairing with the
fundamental projector of the -algebra (the nontrivial generator of the
-group) as well as the pairing with the -analogue of the Bott
projector. Finally, we show that the local index formula is trivially
satisfied.Comment: 18 pages, no figures; minor correction