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Contractions of Poisson-Lie groups are introduced by using Lie bialgebra contractions. As an application, contractions of SL(2,R) Poisson-Lie groups leading to (1+1) Poincaré and Heisenberg structures are analysed. It is shown how the method here introduced allows a systematic construction of the Poisson structures associated to non-coboundary Lie bialgebras. Finally, it is sketched how contractions are also implemented after quantization by using the Lie bialgebra approach.

Creation and annihilation operators for orthogonal polynomials of continuous and discrete variables.

Lorente, Miguel (1999)

ETNA. Electronic Transactions on Numerical Analysis [electronic only]

Crystal bases for the quantum queer superalgebra

Dimitar Grantcharov, Ji Hye Jung, Seok-Jin Kang, Masaki Kashiwara, Myungho Kim (2015)

Journal of the European Mathematical Society

In this paper, we develop the crystal basis theory for the quantum queer superalgebra $U_{q} (𝔮 (n))$ . We define the notion of crystal bases and prove the tensor product rule for $U_{q} (𝔮 (n))$ -modules in the category $𝒪_{int}^{\geq 0}$ . Our main theorem shows that every $U_{q} (𝔮 (n))$ -module in the category $𝒪_{int}^{\geq 0}$ has a unique crystal basis.

Deformation on phase space.

Oscar Arratia, M.ª Angeles Martín Mínguez, María Angeles del Olmo (2002)

RACSAM

El trabajo que presentamos constituye una revisión de varios procedimientos de cuantización basados en un espacio de fases clásico M. Estos métodos consideran a la mecánica cuántica como una "deformación" de la mecánica clásica por medio de la "transformación" del álgebra conmutativa C∞(M) en una nueva álgebra no conmutativa C∞(M)ħ. Todas estas ideas conducen de modo natural a los grupos cuánticos como deformación (o cuantización en un sentido amplio) de los grupos de Poisson-Lie, lo cual también...

Deformed commutators on comodule algebras over coquasitriangular Hopf algebras

Zhongwei Wang, Guoyin Zhang, Liangyun Zhang (2015)

Colloquium Mathematicae

We construct quantum commutators on comodule algebras over coquasitriangular Hopf algebras, so that they are quantum group coinvariant and have the generalized antisymmetry and Leibniz properties. If the coquasitriangular Hopf algebra is additionally cotriangular, then the quantum commutators satisfy a generalized Jacobi identity, and turn the comodule algebra into a quantum Lie algebra. Moreover, we investigate the projective and injective dimensions of some Doi-Hopf modules over a quantum commutative...

Deformed oscillators with two double (pairwise) degeneracies of energy levels.

Gavrilik, Alexandre M., Rebesh, Anastasiya P. (2007)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

Degenerate series representations of the $q$ -deformed algebra ${so}_{q}^{'} (r, s)$ .

Groza, Valentyna A. (2007)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

Determinantal representation of the time-dependent stationary correlation function for the totally asymmetric simple exclusion model.

Bogoliubov, Nikolay M. (2009)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

Differential calculus on 'non-standard' (h-deformed) Minkowski spaces

José de Azcárraga, Francisco Rodenas (1997)

Banach Center Publications

The differential calculus on 'non-standard' h-Minkowski spaces is given. In particular it is shown that, for them, it is possible to introduce coordinates and derivatives which are simultaneously hermitian.

Differential representations of dynamical oscillator symmetries in discrete Hilbert space.

Ruffing, Andreas (2000)

Discrete Dynamics in Nature and Society

Dilogarithm identities for sine-Gordon and reduced sine-Gordon Y-systems.

Nakanishi, Tomoki, Tateo, Roberto (2010)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

Dressing transformation on quantum compact groups

Jurčo, B., Šťovíček, P. (1993)

Proceedings of the Winter School "Geometry and Topology"

Dynamical $ℛ$ matrices of elliptic quantum groups and connection matrices for the $q$ -KZ equations.

Konno, Hitoshi (2006)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

Exponentiations over the quantum algebra $U_{q} (s l_{2} (ℂ))$

Sonia L’Innocente, Françoise Point, Carlo Toffalori (2013)

Confluentes Mathematici

We define and compare, by model-theoretical methods, some exponentiations over the quantum algebra $U_{q} (s l_{2} (ℂ))$ . We discuss two cases, according to whether the parameter $q$ is a root of unity. We show that the universal enveloping algebra of $s l_{2} (ℂ)$ embeds in a non-principal ultraproduct of $U_{q} (s l_{2} (ℂ))$ , where $q$ varies over the primitive roots of unity.

Formula for unbiased bases

Maurice R. Kibler (2010)

Kybernetika

The present paper deals with mutually unbiased bases for systems of qudits in $d$ dimensions. Such bases are of considerable interest in quantum information. A formula for deriving a complete set of $1 + p$ mutually unbiased bases is given for $d = p$ where $p$ is a prime integer. The formula follows from a nonstandard approach to the representation theory of the group $S U (2)$ . A particular case of the formula is derived from the introduction of a phase operator associated with a generalized oscillator algebra. The case...

Free dynamical quantum groups and the dynamical quantum group $S U_{Q}^{d y n} (2)$

Thomas Timmermann (2012)

Banach Center Publications

We introduce dynamical analogues of the free orthogonal and free unitary quantum groups, which are no longer Hopf algebras but Hopf algebroids or quantum groupoids. These objects are constructed on the purely algebraic level and on the level of universal C*-algebras. As an example, we recover the dynamical $S U_{q} (2)$ studied by Koelink and Rosengren, and construct a refinement that includes several interesting limit cases.

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