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Sep 18, 2013
09/13

by
O. Arratia; M. A. del Olmo

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In this paper we show how to construct explicitly induced representations for bicrossproduct Hopf algebras with abelian kernels starting from one-dimensional characters of the commutative sector. We introduce this technique by means of two concrete physical examples: two quantum deformations of the (1+1) Galilei algebra.

Source: http://arxiv.org/abs/math/0110265v1

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29

Sep 18, 2013
09/13

by
O. Arratia; M. A. del Olmo

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We construct the induced representations of the null-plane quantum Poincar\'e and quantum kappa Galilei algebras in (1+1) dimensions. The induction procedure makes use of the concept of module and is based on the existence of a pair of Hopf algebras with a nondegenerate pairing and dual bases.

Source: http://arxiv.org/abs/math/0012227v1

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Sep 21, 2013
09/13

by
O. Arratia; M. A. del Olmo

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We present the q-deformed counterpart of the local representations of the (1+1) extended Galilei group. These representations act on the space of wavefunctions defined in the space-time. As in the classical case the q-local representations are reducible and the condition of irreducibility is given by the q-Casimir equation that in the limit of the deformation parameter going to zero becomes the Schroedinger equation of a free particle.

Source: http://arxiv.org/abs/math/0203039v1

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Sep 22, 2013
09/13

by
E. Celeghini; M. A. del Olmo

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A ladder structure of operators is presented for the associated Legendre polynomials and the spherical harmonics showing that both belong to the same irreducible representation of so(3,2). As both are also bases of square-integrable functions, the universal enveloping algebra of so(3,2) is thus shown to be isomorphic to the space of linear operators acting on the L^2 functions defined on (-1,1) x Z and on the sphere S^2, respectively. The presence of a ladder structure is suggested to be the...

Source: http://arxiv.org/abs/1210.5192v1

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Sep 18, 2013
09/13

by
O. Arratia; M. A. del Olmo

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We analyze the elements characterizing the theory of induced representations of Lie groups, in order to generalize it to quantum groups. We emphasize the geometric and algebraic aspects of the theory, because they are more suitable for generalization in the framework of Hopf algebras. As an example, we present the induced representations of a quantum deformation of the extended Galilei algebra in (1+1) dimensions.

Source: http://arxiv.org/abs/math/0110266v1

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Sep 20, 2013
09/13

by
J. P. Gazeau; M. A. del Olmo

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We revisit the quantized version of the harmonic oscillator obtained through a q-dependent family of coherent states. For each q, 0 < q < 1, these normalized states form an overcomplete set that resolves the unity with respect to an explicit measure. We restrict our study to the case in which 1/q is a quadratic unit Pisot number: the q-deformed integers form Fibonacci-like sequences of integers. We then examine the main characteristics of the corresponding quantum oscillator: localization...

Source: http://arxiv.org/abs/1207.1200v1

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Sep 18, 2013
09/13

by
O. Arratia; M. A. del Olmo

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We present a method to construct induced representations of quantum algebras having the structure of bicrossproduct. We apply this procedure to some quantum kinematical algebras in (1+1)--dimensions with this kind of structure: null-plane quantum Poincare algebra, non-standard quantum Galilei algebra and quantum kappa Galilei algebra.

Source: http://arxiv.org/abs/math/0110275v1

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1.0

Jun 30, 2018
06/18

by
E. Celeghini; M. A. del Olmo

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A ladder algebraic structure for $L^2(\mathbb{R}^+)$ which closes the Lie algebra $h(1)\oplus h(1)$, where $h(1)$ is the Heisenberg-Weyl algebra, is presented in terms of a basis of associated Laguerre polynomials. Using the Schwinger method the quadratic generators that span the alternative Lie algebras $so(3)$, $so(2,1)$ and $so(3,2)$ are also constructed. These families of (pseudo) orthogonal algebras also allow to obtain unitary irreducible representations in $L^2(\mathbb{R}^2)$ similar to...

Topics: Mathematical Physics, Mathematics

Source: http://arxiv.org/abs/1702.02003

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372

Sep 23, 2013
09/13

by
E. Celeghini; M. A. del Olmo

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In Quantum Mechanics operators must be hermitian and, in a direct product space, symmetric. These properties are saved by Lie algebra operators but not by those of quantum algebras. A possible correspondence between observables and quantum algebra operators is suggested by extending the definition of matrix elements of a physical observable, including the eventual projection on the appropriate symmetric space. This allows to build in the Lie space of representations one-parameter families of...

Source: http://arxiv.org/abs/hep-th/0109026v1

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28

Sep 20, 2013
09/13

by
J. A. Calzada; J. Negro; M. A. del Olmo

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A class of two-dimensional superintegrable systems on a constant curvature surface is considered as the natural generalization of some well known one-dimensional factorized systems. By using standard methods to find the shape-invariant intertwining operators we arrive at a $so(6)$ dynamical algebra and its Hamiltonian hierarchies. We pay attention to those associated to certain unitary irreducible representations that can be displayed by means of three-dimensional polyhedral lattices. We also...

Source: http://arxiv.org/abs/math-ph/0601067v1

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Sep 20, 2013
09/13

by
J. A. Calzada; M. A. del Olmo; M. A. Rodriguez

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Integrable systems in low dimensions, constructed through the symmetry reduction method, are studied using phase portrait and variable separation techniques. In particular, invariant quantities and explicit periodic solutions are determined. Widely applied models in Physics are shown to appear as particular cases of the method.

Source: http://arxiv.org/abs/solv-int/9810010v1

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Sep 18, 2013
09/13

by
A. Ballesteros; E. Celeghini; M. A. del Olmo

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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...

Source: http://arxiv.org/abs/math/0611393v1

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0.0

Jun 30, 2018
06/18

by
E. Celeghini; M. A. del Olmo; M. A. Velasco

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A symmetry $SU(2,2)$ group in terms of ladder operators is presented for the Jacobi polynomials, $J_{n}^{(\alpha,\beta)}(x)$, and the Wigner $d_j$-matrices where the spins $j=n+(\alpha+\beta)/2$ integer and half-integer are considered together. A unitary irreducible representation of $SU(2,2)$ is constructed and subgroups of physical interest are discussed. The Universal Enveloping Algebra of $su(2,2)$ also allows to construct group structures $(SU(1,1), SO(3,2), Spin(3,2))$ whose...

Topics: Quantum Physics, Mathematics, Mathematical Physics

Source: http://arxiv.org/abs/1402.5217

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Sep 21, 2013
09/13

by
E. Celeghini; A. Ballesteros; M. A. del Olmo

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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...

Source: http://arxiv.org/abs/0712.0520v1

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Sep 22, 2013
09/13

by
D. Levi; J. Negro; M. A. del Olmo

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In this paper we extend the umbral calculus, developed to deal with difference equations on uniform lattices, to q-difference equations. We show that many of the properties considered for shift invariant difference operators satisfying the umbral calculus can be implemented to the case of the q-difference operators. This q-umbral calculus can be used to provide solutions to linear q-difference equations and q-differential delay equations. To illustrate the method, we will apply the obtained...

Source: http://arxiv.org/abs/math-ph/0310006v1

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Sep 18, 2013
09/13

by
A. Ballesteros; E. Celeghini; M. A. del Olmo

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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...

Source: http://arxiv.org/abs/0803.1426v1

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Sep 23, 2013
09/13

by
A. Ballesteros; E. Celeghini; M. A. del Olmo

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The Poisson-Hopf analogue of an arbitrary quantum algebra U_z(g) is constructed by introducing a one-parameter family of quantizations U_{z,h}(g) depending explicitly on h and by taking the appropriate h -> 0 limit. The q-Poisson analogues of the su(2) algebra are discussed and the novel su_q^P (3) case is introduced. The q-Serre relations are also extended to the Poisson limit. This approach opens the perspective for possible applications of higher rank q-deformed Hopf algebras in...

Source: http://arxiv.org/abs/0903.2178v1

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34

Sep 23, 2013
09/13

by
P. P. Kulish; V. D. Lyakhovsky; M. A. del Olmo

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For chains of regular injections A_p -> A_(p-1) -> ... -> A_1 -> A_0 of Hopf algebras the sets of maximal extended Jordanian twists F_E are considered. We prove that under certain conditions there exists for A_0 the twist composed by the factors (F_E)_k. The general construction of a chain of twists is applied to the universal envelopings U(g) of classical Lie algebras g. We study the chains for the infinite series A_n, B_n and D_n. The properties of the deformation produced by a...

Source: http://arxiv.org/abs/math/9908061v1

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36

Sep 21, 2013
09/13

by
A. Ballesteros; E. Celeghini; M. A. del Olmo

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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...

Source: http://arxiv.org/abs/math/0311146v1

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Sep 18, 2013
09/13

by
M. Gadella; M. A. del Olmo; J. Tosiek

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Some properties of the star product of the Weyl type (i.e. associated with the Weyl ordering) are proved. Fedosov construction of the *-product on a 2-dimensional phase spacewith a constant curvature tensor is presented. Eigenvalue equations for momentum p and position q on a 2-dimensional phase space with constant curvature tensors are solved.

Source: http://arxiv.org/abs/hep-ph/0306117v1

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26

Sep 20, 2013
09/13

by
M. Gadella; M. A. del Olmo; J. Tosiek

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The construction of the *-product proposed by Fedosov is implemented in terms of the theory of fibre bundles. The geometrical origin of the Weyl algebra and the Weyl bundle is shown. Several properties of the product in the Weyl algebra are proved. Symplectic and abelian connections in the Weyl algebra bundle are introduced. Relations between them and the symplectic connection on a phase space M are established. Elements of differential symplectic geometry are included. Examples of the Fedosov...

Source: http://arxiv.org/abs/hep-th/0405157v1

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42

Sep 18, 2013
09/13

by
D. Levi; J. Negro; M. A. del Olmo

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The discrete heat equation is worked out in order to illustrate the search of symmetries of difference equations. It is paid an special attention to the Lie structure of these symmetries, as well as to their dependence on the derivative discretization. The case of q-symmetries for discrete equations in a q-lattice is also considered.

Source: http://arxiv.org/abs/nlin/0012043v1

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Jul 20, 2013
07/13

by
M. A. del Olmo; M. A. Rodriguez; P. Winternitz

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Eleven different types of "maximally superintegrable" Hamiltonian systems on the real hyperboloid $(s^0)^2-(s^1)^2+(s^2)^2-(s^3)^2=1$ are obtained. All of them correspond to a free Hamiltonian system on the homogeneous space $SU(2,2)/U(2,1)$, but to reductions by different maximal abelian subgroups of $SU(2,2)$. Each of the obtained systems allows 5 functionally independent integrals of motion, from which it is possible to form two or more triplets in involution (each of them includes...

Source: http://arxiv.org/abs/hep-th/9407080v1

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5.0

Jun 27, 2018
06/18

by
J. J. Fernandez; J. M. Izquierdo; M. A. del Olmo

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We study here the generalized Weimar-Woods contractions of the superalgebra $osp(1|32) \oplus osp(1|32)$ in order to obtain a suitable algebra that could describe the gauge group of $D=11$ supergravity. The contracted superalgebras are assumed to be given in terms of fermionic extensions of the M-theory superalgebra. We show that the only superalgebra of this type obtained by contraction is the only one for which the three-form of $D=11$ supergravity cannot be trivialized. Therefore, $D=11$...

Topic: High Energy Physics - Theory

Source: http://arxiv.org/abs/1504.05946

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57

Sep 19, 2013
09/13

by
D. Levi; J. Negro; M. A. del Olmo

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We show on the example of the discrete heat equation that for any given discrete derivative we can construct a nontrivial Leibniz rule suitable to find the symmetries of discrete equations. In this way we obtain a symmetry Lie algebra, defined in terms of shift operators, isomorphic to that of the continuous heat equation.

Source: http://arxiv.org/abs/nlin/0010044v1

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Sep 18, 2013
09/13

by
J. Negro; M. A. del Olmo; A. Rodriguez-Marco

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We show that the Landau quantum systems (or integer quantum Hall effect systems) in a plane, sphere or a hyperboloid, can be explained in a complete meaningful way from group-theoretical considerations concerning the symmetry group of the corresponding configuration space. The crucial point in our development is the role played by locality and its appropriate mathematical framework, the fiber bundles. In this way the Landau levels can be understood as the local equivalence classes of the...

Source: http://arxiv.org/abs/quant-ph/0110152v1

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33

Sep 23, 2013
09/13

by
A. Ballesteros; E. Celeghini; M. A. del Olmo

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Hopf algebra quantizations of 4-dimensional and 6-dimensional real classical Drinfel'd doubles are studied by following a direct "analytic" approach. The full quantization is explicitly obtained for most of the Drinfel'd doubles, except a small number of them for which the dual Lie algebra is either sl(2) or so(3). In the latter cases, the classical r-matrices underlying the Drinfel'd double quantizations contain known standard ones plus additional twists. Several new four and...

Source: http://arxiv.org/abs/math/0411389v1

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33

Sep 20, 2013
09/13

by
J. A. Calzada; J. Negro; M. A. del Olmo

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We study the dynamical symmetries of a class of two-dimensional superintegrable systems on a 2-sphere, obtained by a procedure based on the Marsden-Weinstein reduction, by considering its shape-invariant intertwining operators. These are obtained by generalizing the techniques of factorization of one-dimensional systems. We firstly obtain a pair of noncommuting Lie algebras $su(2)$ that originate the algebra $so(4)$. By considering three spherical coordinate systems we get the algebra $u(3)$...

Source: http://arxiv.org/abs/nlin/0601069v1

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Sep 23, 2013
09/13

by
A. Ballesteros; F. J. Herranz; M. A. del Olmo; M. Santander

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A new "non-standard" quantization of the universal enveloping algebra of the split (natural) real form $so(2,2)$ of $D_2$ is presented. Some (classical) graded contractions of $so(2,2)$ associated to a $Z_2 \times Z_2$ grading are studied, and the automorphisms defining this grading are generalized to the quantum case, thus providing quantum contractions of this algebra. This produces a new family of "non-standard" quantum algebras. Some of these algebras can be realized as...

Source: http://arxiv.org/abs/hep-th/9406098v2

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Sep 18, 2013
09/13

by
F. Bonechi; R. Giachetti; M. A. del Olmo; E. Sorace; M. Tarlini

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We study the most elementary aspects of harmonic analysis on a homogeneous space of a deformation of the two-dimensional Euclidean group, admitting generalizations to dimensions three and four, whose quantum parameter has the physical dimensions of a length. The homogeneous space is recognized as a new quantum plane and the action of the Euclidean quantum group is used to determine an eigenvalue problem for the Casimir operator, that constitutes the analogue of the Schroedinger equation in the...

Source: http://arxiv.org/abs/q-alg/9605027v1

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Sep 18, 2013
09/13

by
A. Ballesteros; N. A. Gromov; F. J. Herranz; M. A. del Olmo; M. Santander

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Lie bialgebra contractions are introduced and classified. A non-degenerate coboundary bialgebra structure is implemented into all pseudo-orthogonal algebras $so(p,q)$ starting from the one corresponding to $so(N+1)$. It allows to introduce a set of Lie bialgebra contractions which leads to Lie bialgebras of quasi-orthogonal algebras. This construction is explicitly given for the cases $N=2,3,4$. All Lie bialgebra contractions studied in this paper define Hopf algebra contractions for the...

Source: http://arxiv.org/abs/hep-th/9412083v3

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Sep 20, 2013
09/13

by
A. Ballesteros; F. J. Herranz; M. A. del Olmo; M. Santander

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The N-dimensional Cayley-Klein scheme allows the simultaneous description of $3^N$ geometries (symmetric orthogonal homogeneous spaces) by means of a set of Lie algebras depending on $N$ real parameters. We present here a quantum deformation of the Lie algebras generating the groups of motion of the two and three dimensional Cayley-Klein geometries. This deformation (Hopf algebra structure) is presented in a compact form by using a formalism developed for the case of (quasi) free Lie algebras....

Source: http://arxiv.org/abs/hep-th/9402116v1

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81

Sep 17, 2013
09/13

by
N. Aizawa; F. J. Herranz; J. Negro; M. A. del Olmo

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A new twisted deformation, U_z(so(4,2)), of the conformal algebra of the (3+1)-dimensional Minkowskian spacetime is presented. This construction is provided by a classical r-matrix spanned by ten Weyl-Poincare generators, which generalizes non-standard quantum deformations previously obtained for so(2,2) and so(3,2). However, by introducing a conformal null-plane basis it is found that the twist can indeed be supported by an eight-dimensional carrier subalgebra. By construction the...

Source: http://arxiv.org/abs/hep-th/0207233v1

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Sep 20, 2013
09/13

by
A. Ballesteros; Enrico Celeghini; F. J. Herranz; M. A. del Olmo; M. Santander

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A universal R--matrix for the quantum Heisenberg algebra h(1)q is presented. Despite of the non--quasitriangularity of this Hopf algebra, the quantum group induced from it coincides with the quasitriangular deformation already known.

Source: http://arxiv.org/abs/hep-th/9402127v1

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Sep 18, 2013
09/13

by
J. A. Calzada; S. Kuru; J. Negro; M. A. del Olmo

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A class of quantum superintegrable Hamiltonians defined on a two-dimensional hyperboloid is considered together with a set of intertwining operators connecting them. It is shown that such intertwining operators close a su(2,1) Lie algebra and determine the Hamiltonians through the Casimir operators. By means of discrete symmetries a broader set of operators is obtained closing a so(4,2) algebra. The physical states corresponding to the discrete spectrum of bound states as well as the...

Source: http://arxiv.org/abs/0803.2117v1

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Sep 19, 2013
09/13

by
A. Ballesteros; E. Celeghini; F. J. Herranz; M. A. del Olmo; M. Santander

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A universal quasitriangular $R$--matrix for the non-standard quantum (1+1) Poincar\'e algebra $U_ziso(1,1)$ is deduced by imposing analyticity in the deformation parameter $z$. A family $g_\mu$ of ``quantum graded contractions" of the algebra $U_ziso(1,1)\oplus U_{-z}iso(1,1)$ 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....

Source: http://arxiv.org/abs/q-alg/9501030v1

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Sep 22, 2013
09/13

by
E. Lopez-Sendino; J. Negro; M. A. del Olmo; E. Salgado

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In this paper we present the first steps for obtaining a discrete Quantum Mechanics making use of the Umbral Calculus. The idea is to discretize the continuous Schroedinger equation substituting the continuous derivatives by discrete ones and the space-time continuous variables by well determined operators that verify some Umbral Calculus conditions. In this way we assure that some properties of integrability and symmetries of the continuous equation are preserved and also the solutions of the...

Source: http://arxiv.org/abs/0805.2017v1

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Sep 21, 2013
09/13

by
F. J. Herranz; M. de Montigny; M. A. del Olmo; M. Santander

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We study $\Bbb Z_2^{\otimes N}$ graded contractions of the real compact simple Lie algebra $so(N+1)$, and we identify within them the Cayley-Klein algebras as a naturally distinguished subset.

Source: http://arxiv.org/abs/hep-th/9312126v1

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Sep 19, 2013
09/13

by
A. Ballesteros; F. J. Herranz; M. A. del Olmo; C. M. Pereña; M. Santander

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The Hopf algebra dual form for the non--standard uniparametric deformation of the (1+1) Poincar\'e algebra $iso(1,1)$ is deduced. In this framework, the quantum coordinates that generate $Fun_w(ISO(1,1))$ define an infinite dimensional Lie algebra. A change in the basis of the dual form is obtained in order to compare this deformation to the standard one. Finally, a non--standard quantum Heisenberg group acting on a quantum Galilean plane is obtained.

Source: http://arxiv.org/abs/q-alg/9501029v1