the paper deal with non-commutative geometry. the notion of quantumspheres was introduced by podles. here we define the quantum hermitianmetric on the quantum spaces and find it for the quantum spheres.

The paper deal with non-commutative geometry. The notion of quantumspheres was introduced by podles. Here we define the quantum hermitianmetric on the quantum spaces and find it for the quantum spheres.

We construct wave functions and Dirac operator of spin 1/2 fermions on quantum four-spheres. The construction can be achieved by the q-deformed differential calculus which is manifestly SO(5)q covariant. We evaluate the engenvalue of the Dirac operator on wave functions of the spinors and show that we can define the chiral fermions in such a way that the massless Dirac operator anti-commutes wi...

Operator representations of the cross product ∗-algebra O(S2 qc)⋊Uq(su2) of the Hopf ∗-algebra Uq(su2) and its module ∗-algebras O(S2 qc) of Podles’ spheres are studied. Two classes of representations are described by explicit formulas for the actions of the generators.

We show that the perturbative g invariant of rational homology 3-spheres can be recovered from the LMO invariant for any simple Lie algebra g, i.e., the LMO invariant is universal among the perturbative invariants. This universality was conjectured in [25]. Since the perturbative invariants dominate the quantum invariants of integral homology 3-spheres [13, 14, 15], this implies that the LMO in...

A genuinely three-dimensional system, viz. the hyperbolic four-sphere scattering system, is investigated with classical, semiclassical, and quantum mechanical methods at various center-to-center separations of the spheres. The efficiency and scaling properties of the computations are discussed by comparisons to the two-dimensional three-disk system. While in systems with few degrees of freedom ...

We study covariant differential calculus on the quantum spheres S q . A classification result for covariant first order differential ∗ calculi is proved. As an important step towards a description of the noncommutative geometry of the quantum spheres, a framework of covariant differential calculus is established, including a particular first order calculus obtained by factorization, higher orde...