arXiv subjects
Publications and source records attributed to L. Dabrowski.
In this overview, we study how to reduce the index pairing for a fibre-product C*-algebra to the index pairing for the C*-algebra over which the fibre product is taken. As an example we analyze the case of suspensions and apply it to noncommutative instanton bundles of arbitrary charges over the suspension of quantum deformations of the 3-sphere.
It is shown that the quantum instanton bundle introduced in Commun. Math. Phys. 226, 419-432 (2002) has a bijective canonical map and is, therefore, a coalgebra Galois extension.
We reformulate the concept of connection on a Hopf-Galois extension $B\subseteq P$ in order to apply it in computing the Chern-Connes pairing between the cyclic cohomology $HC^{2n} (B)$ and $K_0 (B)$. This reformulation allows us to show that a Hopf-Galois extension admitting a strong connection is projective and left faithfully flat. It also enables us to conclude that a strong connection is a Cuntz-Quillen-type bimodule connection. To exemplify the theory, we construct a strong connection (super Dirac monopole) to find out the Chern-Connes pairing for the super line bundles associated to super Hopf fibration.
The exact sequence of ``coordinate-ring'' Hopf algebras A(SL(2,C)) -> A(SL_q(2)) -> A(F) determined by the Frobenius map Fr, and the same way obtained exact sequence of (quantum) Borel subgroups, are studied when q is a cubic root of unity. An A(SL(2,C))-linear splitting of A(SL_q(2)) making A(SL(2,C)) a direct summand of A(SL_q(2)) is constructed and used to prove that A(SL_q(2)) is a faithfully flat A(F)-Galois extension of A(SL(2,C)). A cocycle and coaction determining the bicrossed-product structure of the upper-triangular (Borel) quantum subgroup of A(SL_q(2)) are computed explicitly.
A quantum particle interacting with a thin solenoid and a magnetic flux is described by a five-parameter family of Hamilton operators, obtained via the method of self-adjoint extensions. One of the parameters, the value of the flux, corresponds to the Aharonov-Bohm effect; the other four parameters correspond to the strength of a singular potential barrier. The spectrum and eigenstates are computed and the scattering problem is solved.
An isomorphism, up to a twist, between the quasitriangular quantum enveloping algebra U_h(sl(2)) and the (classical) U(sl(2))[[h]]is discussed. The universal twisting element $\cal F$ is given up to the second order in the deformation parameter $h$.
The positive-energy unitary irreducible representations of the $q$-deformed conformal algebra ${\cal C}_q = {\cal U}_q(su(2,2))$ are obtained by appropriate deformation of the classical ones. When the deformation parameter $q$ is $N$-th root of unity, all these unitary representations become finite-dimensional. For this case we discuss in some detail the massless representations, which are also irreducible representations of the $q$-deformed Poincaré subalgebra of ${\cal C}_q$. Generically, their dimensions are smaller than the corresponding finite-dimensional non-unitary representation of $su(2,2)$, except when $N=2$, $h=0$ and $N = 2 \vert h\vert +1$, where $h$ is the helicity of the representations. The latter cases include the fundamental representations with $h = \pm 1/2$.