3,822 research outputs found
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
(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
Bases in Lie and Quantum Algebras
Applications of algebras in physics are related to the connection of
measurable observables to relevant elements of the algebras, usually the
generators. However, in the determination of the generators in Lie algebras
there is place for some arbitrary conventions. The situation is much more
involved in the context of quantum algebras, where inside the quantum universal
enveloping algebra, we have not enough primitive elements that allow for a
privileged set of generators and all basic sets are equivalent. In this paper
we discuss how the Drinfeld double structure underlying every simple Lie
bialgebra characterizes uniquely a particular basis without any freedom,
completing the Cartan program on simple algebras. By means of a perturbative
construction, a distinguished deformed basis (we call it the analytical basis)
is obtained for every quantum group as the analytical prolongation of the above
defined Lie basis of the corresponding Lie bialgebra. It turns out that the
whole construction is unique, so to each quantum universal enveloping algebra
is associated one and only one bialgebra. In this way the problem of the
classification of quantum algebras is moved to the classification of
bialgebras. In order to make this procedure more clear, we discuss in detail
the simple cases of su(2) and su_q(2).Comment: 16 pages, Proceedings of the 5th International Symposium on Quantum
Theory and Symmetries QTS5 (July 22-28, 2007, Valladolid (Spain)
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
Universal --matrices for non-standard (1+1) quantum groups
A universal quasitriangular --matrix for the non-standard quantum (1+1)
Poincar\'e algebra is deduced by imposing analyticity in the
deformation parameter . A family of ``quantum graded contractions"
of the algebra is obtained; this set of
quantum algebras contains as Hopf subalgebras with two primitive translations
quantum analogues of the two dimensional Euclidean, Poincar\'e and Galilei
algebras enlarged with dilations. Universal --matrices
for these quantum Weyl algebras and their associated quantum groups are
constructed.Comment: 12 pages, LaTeX
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
A systematic construction of completely integrable Hamiltonians from coalgebras
A universal algorithm to construct N-particle (classical and quantum)
completely integrable Hamiltonian systems from representations of coalgebras
with Casimir element is presented. In particular, this construction shows that
quantum deformations can be interpreted as generating structures for integrable
deformations of Hamiltonian systems with coalgebra symmetry. In order to
illustrate this general method, the algebra and the oscillator
algebra are used to derive new classical integrable systems including a
generalization of Gaudin-Calogero systems and oscillator chains. Quantum
deformations are then used to obtain some explicit integrable deformations of
the previous long-range interacting systems and a (non-coboundary) deformation
of the Poincar\'e algebra is shown to provide a new
Ruijsenaars-Schneider-like Hamiltonian.Comment: 26 pages, LaTe
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
Universal R-matrix for null-plane quantized Poincar{\'e} algebra
The universal --matrix for a quantized Poincar{\'e} algebra introduced by Ballesteros et al is evaluated. The solution is obtained
as a specific case of a formulated multidimensional generalization to the
non-standard (Jordanian) quantization of .Comment: 9 pages, LaTeX, no figures. The example on page 5 has been
supplemented with the full descriptio
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