arXiv subjects

Angel Ballesteros

Publications and source records attributed to Angel Ballesteros.

119 records · Page 7Linked to original sources

arXiv1998

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 $so(2,1)$ algebra and the oscillator algebra $h_4$ 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 $(1+1)$ Poincaré algebra is shown to provide a new Ruijsenaars-Schneider-like Hamiltonian.

solv-int↗

arXiv1997

Quantum two-photon algebra from non-standard U_z(sl(2,R)) and a discrete time Schrödinger equation

The non-standard quantum deformation of the (trivially) extended sl(2,R) algebra is used to construct a new quantum deformation of the two-photon algebra h_6 and its associated quantum universal R-matrix. A deformed one-boson representation for this algebra is deduced and applied to construct a first order deformation of the differential equation that generates the two-photon algebra eigenstates in Quantum Optics. On the other hand, the isomorphism between h_6 and the (1+1) Schrödinger algebra leads to a new quantum deformation for the latter for which a differential-difference realization is presented. From it, a time discretization of the heat-Schrödinger equation is obtained and the quantum Schrödinger generators are shown to be symmetry operators.

q-alg↗

arXiv1997

A Jordanian quantum two-photon/Schrodinger algebra

A non-standard quantum deformation of the two-photon algebra $h_6$ is constructed, and its quantum universal R-matrix is given. Representations of this new quantum algebra are studied on the Fock space and translated into Fock-Bargmann realizations that provide a direct formalism for the definition of deformed states of light. Finally, the isomorphism between $h_6$ and the (1+1) Schrödinger algebra is used to introduce a new (non-standard) Hopf algebra deformation of this latter symmetry algebra.

q-alg↗

arXiv1997

Contractions on the Classical Double

Lie algebra contractions on the classical Drinfel'd Double of a given Lie bialgebra are introduced and compared to the usual Lie bialgebra contraction theory. The connection between both approaches turns out to be intimately linked to duality problems. The non-relativistic (Galilean) limit of a (1+1) Poincaré Double is used to illustrate the contraction process. Finally, it is shown that, in a certain sense, the classical limit in a quantum algebra can be thought as a certain contraction on the corresponding Double.

q-alg↗

arXiv1997

Harmonic Oscillator Lie Bialgebras and their Quantization

All possible Lie bialgebra structures on the harmonic oscillator algebra are explicitly derived and it is shown that all of them are of the coboundary type. A non-standard quantum oscillator is introduced as a quantization of a triangular Lie bialgebra, and a universal $R$-matrix linked to this new quantum algebra is presented.

q-alg↗

arXiv1997

Boson representations, non-standard quantum algebras and contractions

A Gelfan'd--Dyson mapping is used to generate a one-boson realization for the non-standard quantum deformation of $sl(2,\R)$ which directly provides its infinite and finite dimensional irreducible representations. Tensor product decompositions are worked out for some examples. Relations between contraction methods and boson realizations are also explored in several contexts. So, a class of two-boson representations for the non-standard deformation of $sl(2,\R)$ is introduced and contracted to the non-standard quantum (1+1) Poincaré representations. Likewise, a quantum extended Hopf $sl(2,\R)$ algebra is constructed and the Jordanian $q$-oscillator algebra representations are obtained from it by means of another contraction procedure.

q-alg↗

arXiv1996

Quantum Heisenberg--Weyl Algebras

All Lie bialgebra structures on the Heisenberg--Weyl algebra $[A_+,A_-]=M$ are classified and explicitly quantized. The complete list of quantum Heisenberg--Weyl algebras so obtained includes new multiparameter deformations, most of them being of the non-coboundary type.

q-alg↗

arXiv1996

Lie bialgebra quantizations of the oscillator algebra and their universal $R$--matrices

All coboundary Lie bialgebras and their corresponding Poisson--Lie structures are constructed for the oscillator algebra generated by $\{å,\ap,\am,\bb\}$. Quantum oscillator algebras are derived from these bialgebras by using the Lyakhovsky and Mudrov formalism and, for some cases, quantizations at both algebra and group levels are obtained, including their universal $R$--matrices.

q-alg↗

arXiv1996

Universal $R$-matrix for non-standard quantum $sl(2,\R)$

A universal $R$-matrix for the non-standard (Jordanian) quantum deformation of $sl(2,\R)$ is presented. A family of solutions of the quantum Yang--Baxter equation is obtained from some finite dimensional representations of this Lie bialgebra quantization of $sl(2,\R)$.

q-alg↗

arXiv1996

(2+1) null-plane quantum Poincaré group from a factorized universal $R$-matrix

The non-standard (Jordanian) quantum deformations of $so(2,2)$ and (2+1) Poincaré algebras are constructed by starting from a quantum $sl(2,\R)$ basis such that simple factorized expressions for their corresponding universal $R$-matrices are obtained. As an application, the null-plane quantum (2+1) Poincaré Poisson-Lie group is quantized by following the FRT prescription. Matrix and differential representations of this null-plane deformation are presented, and the influence of the choice of the basis in the resultant $q$-Schrödinger equation governing the deformed null plane evolution is commented.

q-alg↗

arXiv1994

Multiboson Expansions for the q-Oscillator and $SU(1,1)_q$

All the hermitian representations of the ``symmetric" $q$-oscillator are obtained by means of expansions. The same technique is applied to characterize in a systematic way the $k$-order boson realizations of the $q$-oscillator and $su(1,1)_q$. The special role played by the quadratic realizations of $su(1,1)_q$ in terms of boson and $q$-boson operators is analysed and clarified.

hep-th↗