20 research outputs found
The Gervais-Neveu-Felder equation for the Jordanian quasi-Hopf U_{h;y}(sl(2)) algebra
Using a contraction procedure, we construct a twist operator that satisfies a
shifted cocycle condition, and leads to the Jordanian quasi-Hopf U_{h;y}(sl(2))
algebra. The corresponding universal matrix obeys a
Gervais-Neveu-Felder equation associated with the U_{h;y}(sl(2)) algebra. For a
class of representations, the dynamical Yang-Baxter equation may be expressed
as a compatibility condition for the algebra of the Lax operators.Comment: Latex, 9 pages, no figure
h-deformation of GL(1|1)
h-deformation of (graded) Hopf algebra of functions on supergroup GL(1|1) is
introduced via a contration of GL_q (1|1). The deformation parameter h is odd
(grassmann). Related differential calculus on h-superplane is presented.Comment: latex file, 8 pages, minor change
Irreducible decomposition for tensor prodect representations of Jordanian quantum algebras
Tensor products of irreducible representations of the Jordanian quantum
algebras U_h(sl(2)) and U_h(su(1,1)) are considered. For both the highest
weight finite dimensional representations of U_h(sl(2)) and lowest weight
infinite dimensional ones of U_h(su(1,1)), it is shown that tensor product
representations are reducible and that the decomposition rules to irreducible
representations are exactly the same as those of corresponding Lie algebras.Comment: LaTeX, 14pages, no figur
Degenerate Sklyanin Algebras
New trigonometric and rational solutions of the quantum Yang-Baxter equation
(QYBE) are obtained by applying some singular gauge transformations to the
known Belavin-Drinfeld elliptic R-matrix for . These
solutions are shown to be related to the standard ones by the quasi-Hopf twist.
We demonstrate that the quantum algebras arising from these new R-matrices can
be obtained as special limits of the Sklyanin algebra. A representation for
these algebras by the difference operators is found. The
-case is discussed.Comment: 12 page
Three dimensional quantum algebras: a Cartan-like point of view
A perturbative quantization procedure for Lie bialgebras is introduced and
used to classify all three dimensional complex quantum algebras compatible with
a given coproduct. The role of elements of the quantum universal enveloping
algebra that, analogously to generators in Lie algebras, have a distinguished
type of coproduct is discussed, and the relevance of a symmetrical basis in the
universal enveloping algebra stressed. New quantizations of three dimensional
solvable algebras, relevant for possible physical applications for their
simplicity, are obtained and all already known related results recovered. Our
results give a quantization of all existing three dimensional Lie algebras and
reproduce, in the classical limit, the most relevant sector of the complete
classification for real three dimensional Lie bialgebra structures given by X.
Gomez in J. Math. Phys. Vol. 41. (2000) 4939.Comment: LaTeX, 15 page
Tensor Operators for Uh(sl(2))
Tensor operators for the Jordanian quantum algebra Uh(sl(2)) are considered.
Some explicit examples of them, which are obtained in the boson or fermion
realization, are given and their properties are studied. It is also shown that
the Wigner-Eckart's theorem can be extended to Uh(sl(2)).Comment: 11pages, LaTeX, to be published in J. Phys.
Jordanian Quantum Algebra via Contraction Method and Mapping
Using the contraction procedure introduced by us in Ref. \cite{ACC2}, we
construct, in the first part of the present letter, the Jordanian quantum Hopf
algebra which has a remarkably simple coalgebraic
structure and contains the Jordanian Hopf algebra ,
obtained by Ohn, as a subalgebra. A nonlinear map between and the classical algebra is then established. In the second
part, we give the higher dimensional Jordanian algebras for all . The Universal -matrix of is also given.Comment: 17 pages, Late
Superintegrability on sl(2)-coalgebra spaces
We review a recently introduced set of N-dimensional quasi-maximally
superintegrable Hamiltonian systems describing geodesic motions, that can be
used to generate "dynamically" a large family of curved spaces. From an
algebraic viewpoint, such spaces are obtained through kinetic energy
Hamiltonians defined on either the sl(2) Poisson coalgebra or a quantum
deformation of it. Certain potentials on these spaces and endowed with the same
underlying coalgebra symmetry have been also introduced in such a way that the
superintegrability properties of the full system are preserved. Several new N=2
examples of this construction are explicitly given, and specific Hamiltonians
leading to spaces of non-constant curvature are emphasized.Comment: 12 pages. Based on the contribution presented at the "XII
International Conference on Symmetry Methods in Physics", Yerevan (Armenia),
July 2006. To appear in Physics of Atomic Nucle
Maps between Deformed and Ordinary Gauge Fields
In this paper, we introduce a map between the q-deformed gauge fields defined
on the GL-covariant quantum hyperplane and the ordinary gauge fields.
Perturbative analysis of the q-deformed QED at the classical level is presented
and gauge fixing la BRST is discussed. An other star product
defined on the hybrid % -plane is explicitly constructed .Comment: 10 page
A unified approach to Poisson-Hopf deformations of Lie-Hamilton systems based on sl(2)
Producción CientíficaBased on a recently developed procedure to construct Poisson-Hopf deformations of Lie–Hamilton systems, a novel unified approach to nonequivalent deformations of Lie–Hamilton systems on the real plane with a Vessiot–Guldberg Lie algebra isomorphic to sl(2) is proposed. This, in particular, allows us to define a notion of Poisson–Hopf systems in dependence of a
parameterized family of Poisson algebra representations. Such an approach is explicitly illustrated by applying it to the three non-diffeomorphic classes of sl(2) Lie–Hamilton systems. Our results cover deformations of the Ermakov system, Milne–Pinney, Kummer–Schwarz and several Riccati equations as well as of the harmonic oscillator (all of them with t-dependent coefficients). Furthermore t-independent constants of motion are given as well. Our methods can be employed to generate other Lie–Hamilton systems and their deformations for other Vessiot–Guldberg Lie algebras and their deformations