942 research outputs found
Realisations of quantum group and its coloured extension through a novel Hopf algebra with five generators
A novel Hopf algebra , depending on two deformation
parameters and five generators, has been constructed. This
Hopf algebra might be considered as some quantisation of classical group, which contains the standard quantum group
(with ) as a Hopf subalgebra. However, we interestingly observe
that the two parameter deformed quantum group can also be
realised through the generators of this algebra, provided
the sets of deformation parameters and are related to each other
in a particular fashion. Subsequently we construct the invariant noncommutative
planes associated with algebra and show how the two well
known Manin planes corresponding to quantum group can easily be
reproduced through such construction. Finally we consider the `coloured'
extension of quantum group as well as corresponding Manin planes
and explore their intimate connection with the `coloured' extension of Hopf structure.Comment: 24 page
Fractional statistics in some exactly solvable Calogero-like models with PT invariant interactions
Here we review a method for constructing exact eigenvalues and eigenfunctions
of a many-particle quantum system, which is obtained by adding some
nonhermitian but PT invariant (i.e., combined parity and time reversal
invariant) interaction to the Calogero model. It is shown that such extended
Calogero model leads to a real spectrum obeying generalised exclusion
statistics. It is also found that the corresponding exchange statistics
parameter differs from the exclusion statistics parameter and exhibits a
`reflection symmetry' provided the strength of the PT invariant interaction
exceeds a critical value.Comment: 8 pages, Latex, Talk given at Joint APCTP-Nankai Symposium, Tianjin
(China), Oct. 200
Algebraic aspect and construction of Lax operators in quantum integrable systems
An algebraic construction more general and intimately connected with that of
Faddeev, along with its application for generating different classes of
quantum integrable models are summarised to complement the recent results of
ref. 1 ( L.D. Faddeev, {\it Int. J. Mod. Phys. } {\bf A10}, 1845 (1995) ).Comment: 8 pages, plain TEX, no figure
Super Rogers-Szeg\"o polynomials associated with type of Polychronakos spin chains
As is well known, multivariate Rogers-Szeg\"o polynomials are closely
connected with the partition functions of the type of Polychronakos
spin chains having long-range interactions. Applying the `freezing trick', here
we derive the partition functions for a class of type of Polychronakos
spin chains containing supersymmetric analogues of polarized spin reversal
operators and subsequently use those partition functions to obtain novel
multivariate super Rogers-Szeg\"o (SRS) polynomials depending on four types of
variables. We construct the generating functions for such SRS polynomials and
show that these polynomials can be written as some bilinear combinations of the
type of SRS polynomials. We also use the above mentioned generating
functions to derive a set of recursion relations for the partition functions of
the type of Polychronakos spin chains involving different numbers of
lattice sites and internal degrees of freedom.Comment: 33 pages, minor typos corrected, journal reference give
Low energy properties of the SU(m|n) supersymmetric Haldane-Shastry spin chain
The ground state and low energy excitations of the SU(m|n) supersymmetric
Haldane-Shastry spin chain are analyzed. In the thermodynamic limit, it is
found that the ground state degeneracy is finite only for the SU(m|0) and
SU(m|1) spin chains, while the dispersion relation for the low energy and low
momentum excitations is linear for all values of m and n. We show that the low
energy excitations of the SU(m|1) spin chain are described by a conformal field
theory of m non-interacting Dirac fermions which have only positive energies;
the central charge of this theory is m/2. Finally, for n \ge 1, the partition
functions of the SU(m|n) Haldane-Shastry spin chain and the SU(m|n)
Polychronakos spin chain are shown to be related in a simple way in the
thermodynamic limit at low temperatures.Comment: 40 pages including 2 figures; added some references; this version
will appear in Nuclear Physics
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