3,325 research outputs found
(1+1) Schrodinger Lie bialgebras and their Poisson-Lie groups
All Lie bialgebra structures for the (1+1)-dimensional centrally extended
Schrodinger algebra are explicitly derived and proved to be of the coboundary
type. Therefore, since all of them come from a classical r-matrix, the complete
family of Schrodinger Poisson-Lie groups can be deduced by means of the
Sklyanin bracket. All possible embeddings of the harmonic oscillator, extended
Galilei and gl(2) Lie bialgebras within the Schrodinger classification are
studied. As an application, new quantum (Hopf algebra) deformations of the
Schrodinger algebra, including their corresponding quantum universal
R-matrices, are constructed.Comment: 25 pages, LaTeX. Possible applications in relation with integrable
systems are pointed; new references adde
Classical Dynamical Systems from q-algebras:"cluster" variables and explicit solutions
A general procedure to get the explicit solution of the equations of motion
for N-body classical Hamiltonian systems equipped with coalgebra symmetry is
introduced by defining a set of appropriate collective variables which are
based on the iterations of the coproduct map on the generators of the algebra.
In this way several examples of N-body dynamical systems obtained from
q-Poisson algebras are explicitly solved: the q-deformed version of the sl(2)
Calogero-Gaudin system (q-CG), a q-Poincare' Gaudin system and a system of
Ruijsenaars type arising from the same (non co-boundary) q-deformation of the
(1+1) Poincare' algebra. Also, a unified interpretation of all these systems as
different Poisson-Lie dynamics on the same three dimensional solvable Lie group
is given.Comment: 19 Latex pages, No figure
Binary trees, coproducts, and integrable systems
We provide a unified framework for the treatment of special integrable
systems which we propose to call "generalized mean field systems". Thereby
previous results on integrable classical and quantum systems are generalized.
Following Ballesteros and Ragnisco, the framework consists of a unital algebra
with brackets, a Casimir element, and a coproduct which can be lifted to higher
tensor products. The coupling scheme of the iterated tensor product is encoded
in a binary tree. The theory is exemplified by the case of a spin octahedron.Comment: 15 pages, 6 figures, v2: minor correction in theorem 1, two new
appendices adde
Non-coboundary Poisson-Lie structures on the book group
All possible Poisson-Lie (PL) structures on the 3D real Lie group generated
by a dilation and two commuting translations are obtained. Its classification
is fully performed by relating these PL groups with the corresponding Lie
bialgebra structures on the corresponding "book" Lie algebra. By construction,
all these Poisson structures are quadratic Poisson-Hopf algebras for which the
group multiplication is a Poisson map. In contrast to the case of simple Lie
groups, it turns out that most of the PL structures on the book group are
non-coboundary ones. Moreover, from the viewpoint of Poisson dynamics, the most
interesting PL book structures are just some of these non-coboundaries, which
are explicitly analysed. In particular, we show that the two different
q-deformed Poisson versions of the sl(2,R) algebra appear as two distinguished
cases in this classification, as well as the quadratic Poisson structure that
underlies the integrability of a large class of 3D Lotka-Volterra equations.
Finally, the quantization problem for these PL groups is sketched.Comment: 15 pages, revised version, some references adde
The spin 1/2 Calogero-Gaudin System and its q-Deformation
The spin 1/2 Calogero-Gaudin system and its q-deformation are exactly solved:
a complete set of commuting observables is diagonalized, and the corresponding
eigenvectors and eigenvalues are explicitly calculated. The method of solution
is purely algebraic and relies on the co-algebra simmetry of the model.Comment: 15 page
Sheffield University CLEF 2000 submission - bilingual track: German to English
We investigated dictionary based cross language information
retrieval using lexical triangulation. Lexical triangulation combines the results
of different transitive translations. Transitive translation uses a pivot language
to translate between two languages when no direct translation resource is
available. We took German queries and translated then via Spanish, or Dutch
into English. We compared the results of retrieval experiments using these
queries, with other versions created by combining the transitive translations or
created by direct translation. Direct dictionary translation of a query introduces
considerable ambiguity that damages retrieval, an average precision 79% below
monolingual in this research. Transitive translation introduces more ambiguity,
giving results worse than 88% below direct translation. We have shown that
lexical triangulation between two transitive translations can eliminate much of
the additional ambiguity introduced by transitive translation
Classical Lie algebras and Drinfeld doubles
The Drinfeld double structure underlying the Cartan series An, Bn, Cn, Dn of
simple Lie algebras is discussed.
This structure is determined by two disjoint solvable subalgebras matched by
a pairing. For the two nilpotent positive and negative root subalgebras the
pairing is natural and in the Cartan subalgebra is defined with the help of a
central extension of the algebra.
A new completely determined basis is found from the compatibility conditions
in the double and a different perspective for quantization is presented. Other
related Drinfeld doubles on C are also considered.Comment: 11 pages. submitted for publication to J. Physics
Multiparametric quantum gl(2): Lie bialgebras, quantum R-matrices and non-relativistic limits
Multiparametric quantum deformations of are studied through a
complete classification of Lie bialgebra structures. From them, the
non-relativistic limit leading to harmonic oscillator Lie bialgebras is
implemented by means of a contraction procedure. New quantum deformations of
together with their associated quantum -matrices are obtained and
other known quantizations are recovered and classified. Several connections
with integrable models are outlined.Comment: 21 pages, LaTeX. To appear in J. Phys. A. New contents adde
Antiferromagnetic O(N) models in four dimensions
We study the antiferromagnetic O(N) model in the F_4 lattice. Monte Carlo
simulations are applied for investigating the behavior of the transition for
N=2,3. The numerical results show a first order nature but with a large
correlation length. The limit is also considered with analytical
methods.Comment: 14 pages, 3 postscript figure
On the biparametric quantum deformation of GL(2) x GL(1)
We study the biparametric quantum deformation of GL(2) x GL(1) and exhibit
its cross-product structure. We derive explictly the associated dual algebra,
i.e., the quantised universal enveloping algebra employing the R-matrix
procedure. This facilitates construction of a bicovariant differential calculus
which is also shown to have a cross-product structure. Finally, a Jordanian
analogue of the deformation is presented as a cross-product algebra.Comment: 16 pages LaTeX, published in JM
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