NONCOMMUTATIVE GEOMETRY, POISSON GEOMETRY AND QUANTUM GROUPS


The aim of this research project is mainly the study of the
theory and the applications of quantum groups and
noncommutative geometry with Poisson geometry as semi
classical limit; algebraic methods in quantum field theory
are also investigated.
This project started in the 1991 and by then it covered many
aspects of the developments of the field. We give a summary
of the subjects treated: contractions of quantum groups,
tensor operators and q-oscillators, applications to the
rotational spectra of heavy nuclei, quantum Galilei group
and the XXZ Heisenberg magnetic model, deformation of
exceptional superalgebras in term of q-parafermions, the
kappa deformed Dirac operator, the exponential mapping for
non semisimple quantum groups, quantum groups coherent
states and squeezed states in lattice quantum mechanics,
q-difference realization of quantum algebras, deformation
quantization of the Heisenberg group, homogeneous spaces of
Euclidean quantum groups, free q-Schro"dinger equation,
lattice space time and k-Poincare', q-orthogonal and
q-ultraspherical polynomials, Drinfeld quantum double and
cohomological properties of the noncommutative differential
calculi, anyonic realization of the quantum affine Lie
algebras and superalgebras, algebraic interpretation of
q-Hermite polynomials and q-Jacobi polynomial, symmetries
and continuous q-orthogonal polynomials, induced
representation of the quantum groups, study of the quantum
deformations of the maximal subalgebras of unitary groups.

During the last years the research has been devoted to the
following topics:
Quantum homogeneous spaces from coisotropic subgroups.
The unitarity of the induced representations from
coisotropic quantum groups.
Description of the many particles systems and of their
statistical properties in terms of Hopf coalgebras.
Deformation of the osp(1|2) algebra using the technique of
the Drinfel'd twist.
Description of deformed simple algebras as quantum doubles.
Analytical classification of low dimension quantum algebras.
The quantum algebra U_q(sl(2)+sl(2)), in the crystal bases,
as the symmetry of the genetic code dynamics.
Neutrinos mixing and oscillation.
Noncommutitative instantons from quantum groups; the quantum
deformation of the S^7 -> S^4 principal fiber bundle.
The Chern-Connes pairing between some Fredholm modules and
associated fiber bundles for the non commutative instanton.
Vertex operator realization of Lorentzian algebras and
Kac-Moody hyperbolic superalgebras.
Field theories of scalar fields or antisymmetric tensors in
R_q^N.
The problem of the cyclic property is the integration over
R_q^N and the formulation of gauge theories.



SUMMARY OF THE RECENT RESULTS OF
THE FLORENCE GROUP

The noncommutative geometries of the homogeneous spaces
obtained from the quantum groups have been studied in the
case of SL(2,R).
We have stressed the importance of the concept of
coisotropic subgroup both at the semi classical and quantum
level in recognizing the homogeneous noncommutative spaces.
The interest for solitons and instantons in noncommutative
geometry has driven us to find the noncommutative version of
the usual principal fiber bundle S^7 over S^4, the
noncommutative S^4 we obtained has been generalized to
quantum even dimensional spheres S^2n, we proved that the
quantum instanton bundle has a bijective canonical
map,therefore it is a coalgebra Galois extension.Our
construction gives the first completely nontrivial example
of noncommutative principal bundle.

It has been studied the Poisson sigma model, it can be
viewed as a topological string theory. Mainly we concentrate
our attention on the Poisson sigma model over a group
manifold G with a Poisson-Lie structure. In this case the
flat connection conditions arise naturally.
The boundary conditions (D-branes) are studied in this
model. It turns out that the D-branes are labelled by the
coisotropic subgroups of G.
We give a description of the moduli space of classical
solutions over Riemann surfaces both without and with
boundaries.

We have considered the relativistic quantum mechanics of a
two interacting fermions system. We quantized the system
with a general interaction potential and gave the explicit
equations in a spherical basis. The case of the Coulomb
interaction is studied in detail by numerical methods,
solving the eigenvalue problem and determining the spectral
curves for a varying ratio of the mass of the interacting
particles.

SUMMARY OF THE RECENT RESULTS OF
THE NAPLES GROUP

In the framework of the “crystal basis model” of genetic
code based on U_{q ->0} (sl(2)+sl(2)): a correlation
between the codon usage frequencies has been put into
evidence which naturally fits in the model; the affinity of
the physico-chemical properties in different amino-acids
have been put into relation with the vicinity of
the encoding codons in the space of the different
irreducible representations which codons belong to;
requiring stability of genetic code against misreading,
modellised by suitable crystal tensor operators, the number
and the structure of the different multiplets have
been reasonably explained; a sum rule for the usage
probability of codons, belongig to sextets and quartets,
with C (cytosine) and A (adenine) as third nucleotiode, has
been derived and satisfactorily compared with data for
vertebrates; the codons distribution functions,
for biological species with high statistics, have been best
fitted by a three parameters expression, sum of an
exponential, a linear terma and a costant.

A preliminary result, in the interesting and ambitious
program of developing (quantum) field theory on the
SO_q(N)-covariant quantum Euclidean spaces R_q^N,
multi(anti)instanton solutions of the (anti)selfduality
equations for a su(2)-type Yang-Mills potential has
been found. Moreover GL_q(N) and SO_q(N)-covariant
deformations of the completely symmetric /antisymmetric
has been obtained.
Adopting the GL_q(N) and SO_q(N)-covariant differential
calculi on the corresponding quantum group covariant
noncommutative spaces, a generalized notion of vielbein
basis has been obtained. A thorough definition of a
SO_q(N)-covariant R_q^N bilinear Hodge map
has been given, acting on the bimodule of differential
forms, introducing the exterior coderivative and it has
been shown that the Laplacian acts on differential forms
exactly as in the undeformed case.
The conventional definition of scalar product in the spaces
of of functions and differential k-forms on R_q^N has been
modified ,by introducing suitable ``weights'' in the
integrals, namely scalar positive-definite
q-pseudodifferential operators, making the exterior
derivative and coderivate hermitean conjugate of each other.


SUMMARY OF THE RECENT RESULTS OF
THE SALERNO GROUP

The activity of the Salerno Group in the frame of the INFN
Project FI42 has been focused in recent years on the study
of the q-deformed Hopf algebra and of group contraction in
connection with several applications of physical interest,
ranging, in a satisfactory unifying perspective, from
quantum dissipative systems to field mixing and to field
quantization in a curved background. The aim has been and is
to uncover the physical content of some of the involved
formal features. In this respect, a crucial finding has been
the recognition of the strict relation between the angle
parameter of the Bogoliubov transformations in quantum field
theory (for fermions as well as for bosons) and the
q-deformation parameter. This result has opened the way to
the possibility of parametrizing the unitarily inequivalent
representations of the canonical (anti-)commutation
relations by means of the q-deformation parameter. The
consequence has been a clear picture of the role of the
(deformed) Hopf coproduct mapping in the full space of the
representations and the possibility to uncover intriguing
non-perturbative aspects in several physical contexts. Among
others, the non-perturbative nature of the flavoured mixed
neutrinos and their exact oscillation formula, of which the
usual Pontecorvo formula turns out to be only the
relativistic limit; the possibility to relate deterministic
classical systems with information loss to the spectrum of
corresponding quantum systems and their thermodynamics, on
the line of the 't Hooft proposal; the discovery of
geometric phases in the mixing physics; the discovery of a
dissipative geometric phase in quantum dissipative systems
and its relation with the zero-point vacuum energy and with
the non-commutative geometric plane; the role of the
Bogoliubov coefficients in the non-zero contribution of the
mixed neutrinos to the value of the cosmological constant;
the possibility of computing the entanglement entropy for
quantum fields near the event horizon. Some of these results
have been confirmed by other working groups. For example,
Fujii in Sapporo, Ji in North Caroline have confirmed and
extended the results on the mixing physics. Hannabus in
Oxford has confirmed the results by producing a
mathematically rigorous proof for any number of neutrinos.
't Hooft has quoted the 2001 paper on dissipation and
quantization. The 1995 paper on mixed fields and the 1992
paper on quantum dissipation are rated as 'well known'
papers in the SLAC-Spires archive.