3,769 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 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
From Quantum Universal Enveloping Algebras to Quantum Algebras
The ``local'' structure of a quantum group G_q is currently considered to be
an infinite-dimensional object: the corresponding quantum universal enveloping
algebra U_q(g), which is a Hopf algebra deformation of the universal enveloping
algebra of a n-dimensional Lie algebra g=Lie(G). However, we show how, by
starting from the generators of the underlying Lie bialgebra (g,\delta), the
analyticity in the deformation parameter(s) allows us to determine in a unique
way a set of n ``almost primitive'' basic objects in U_q(g), that could be
properly called the ``quantum algebra generators''. So, the analytical
prolongation (g_q,\Delta) of the Lie bialgebra (g,\delta) is proposed as the
appropriate local structure of G_q. Besides, as in this way (g,\delta) and
U_q(g) are shown to be in one-to-one correspondence, the classification of
quantum groups is reduced to the classification of Lie bialgebras. The su_q(2)
and su_q(3) cases are explicitly elaborated.Comment: 16 pages, 0 figures, LaTeX fil
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
Bicrossproduct structure of the null-plane quantum Poincare algebra
A nonlinear change of basis allows to show that the non-standard quantum
deformation of the (3+1) Poincare algebra has a bicrossproduct structure.
Quantum universal R-matrix, Pauli-Lubanski and mass operators are presented in
the new basis.Comment: 7 pages, LaTe
Quantum (1+1) extended Galilei algebras: from Lie bialgebras to quantum R-matrices and integrable systems
The Lie bialgebras of the (1+1) extended Galilei algebra are obtained and
classified into four multiparametric families. Their quantum deformations are
obtained, together with the corresponding deformed Casimir operators. For the
coboundary cases quantum universal R-matrices are also given. Applications of
the quantum extended Galilei algebras to classical integrable systems are
explicitly developed.Comment: 16 pages, LaTeX. A detailed description of the construction of
integrable systems is carried ou
Universal integrals for superintegrable systems on N-dimensional spaces of constant curvature
An infinite family of classical superintegrable Hamiltonians defined on the
N-dimensional spherical, Euclidean and hyperbolic spaces are shown to have a
common set of (2N-3) functionally independent constants of the motion. Among
them, two different subsets of N integrals in involution (including the
Hamiltonian) can always be explicitly identified. As particular cases, we
recover in a straightforward way most of the superintegrability properties of
the Smorodinsky-Winternitz and generalized Kepler-Coulomb systems on spaces of
constant curvature and we introduce as well new classes of (quasi-maximally)
superintegrable potentials on these spaces. Results here presented are a
consequence of the sl(2) Poisson coalgebra symmetry of all the Hamiltonians,
together with an appropriate use of the phase spaces associated to Poincare and
Beltrami coordinates.Comment: 12 page
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
Induced Representations of Quantum Kinematical Algebras and Quantum Mechanics
Unitary representations of kinematical symmetry groups of quantum systems are
fundamental in quantum theory. We propose in this paper its generalization to
quantum kinematical groups. Using the method, proposed by us in a recent paper
(olmo01), to induce representations of quantum bicrossproduct algebras we
construct the representations of the family of standard quantum inhomogeneous
algebras . This family contains the quantum
Euclidean, Galilei and Poincar\'e algebras, all of them in (1+1) dimensions. As
byproducts we obtain the actions of these quantum algebras on regular co-spaces
that are an algebraic generalization of the homogeneous spaces and --Casimir
equations which play the role of --Schr\"odinger equations.Comment: LaTeX 2e, 20 page
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