On the quantum group and quantum algebra approach toq-special functions

1993 ◽  
Vol 27 (3) ◽  
pp. 179-190 ◽  
Author(s):  
Roberto Floreanini ◽  
Luc Vinet
Keyword(s):  
Quantum Group ◽  
Special Functions ◽  
Quantum Algebra ◽  
Algebra Approach

A quantum algebra approach to basic multivariable special functions

1994 ◽  
Vol 27 (20) ◽  
pp. 6781-6797 ◽  
Author(s):  
R Floreanini ◽  
L Lapointe ◽  
L Vinet
Keyword(s):  
Special Functions ◽  
Quantum Algebra ◽  
Algebra Approach

scholarly journals WAKIMOTO REALIZATION OF THE ELLIPTIC QUANTUM GROUP $U_{q,p}(\widehat{{\rm sl}_N})$

2009 ◽  
Vol 24 (30) ◽  
pp. 5561-5578
Author(s):  
TAKEO KOJIMA
Keyword(s):  
Quantum Group ◽  
Free Field ◽  
Quantum Algebra ◽  
Free Field Realization ◽  
Wakimoto Realization ◽  
Arbitrary Level ◽  
Elliptic Quantum

We construct a free field realization of the elliptic quantum algebra [Formula: see text] for arbitrary level k ≠ 0, -N. We study Drinfeld current and the screening current associated with [Formula: see text] for arbitrary level k. In the limit p → 0 this realization becomes q-Wakimoto realization for [Formula: see text].

scholarly journals Multivariableq-Hahn polynomials as coupling coefficients for quantum algebra representations

2001 ◽  
Vol 28 (6) ◽  
pp. 331-358 ◽  
Author(s):  
Hjalmar Rosengren
Keyword(s):  
Tensor Product ◽  
Quantum Group ◽  
Quantum Algebra ◽  
Coupling Coefficients ◽  
Highest Weight ◽  
Hahn Polynomials ◽  
Highest Weight Representations

We study coupling coefficients for a multiple tensor product of highest weight representations of theSU(1,1)quantum group. These are multivariable generalizations of theq-Hahn polynomials.

scholarly journals LANDAU LEVELS AND QUANTUM GROUP

1994 ◽  
Vol 09 (05) ◽  
pp. 451-458 ◽  
Author(s):  
HARU-TADA SATO
Keyword(s):  
Quantum Group ◽  
Uniform Magnetic Field ◽  
Periodic Potential ◽  
Group Structure ◽  
Quantum Algebra ◽  
Wave Functions ◽  
Landau Levels ◽  
Group Symmetry ◽  
Factor V ◽  
Level Degeneracy

We find a quantum group structure in two-dimensional motions of a nonrelativistic electron in a uniform magnetic field and in a periodic potential. The representation basis of the quantum algebra is composed of wave functions of the system. The quantum group symmetry commutes with the Hamiltonian and is relevant to the Landau level degeneracy. The deformation parameter q of the quantum algebra turns out to be given by the fractional filling factor v=1/m (m odd integer).

scholarly journals A Quantum Algebra Approach to Multivariate Askey–Wilson Polynomials

2019 ◽  
Author(s):  
Wolter Groenevelt
Keyword(s):  
Quantum Algebra ◽  
Matrix Elements ◽  
Primitive Elements ◽  
Difference Problem ◽  
Change Of Representation ◽  
Algebra Approach ◽  
Orthogonality Relations ◽  
The Matrix ◽  
Wilson Polynomials

Abstract We study matrix elements of a change of basis between two different bases of representations of the quantum algebra ${\mathcal{U}}_q(\mathfrak{s}\mathfrak{u}(1,1))$. The two bases, which are multivariate versions of Al-Salam–Chihara polynomials, are eigenfunctions of iterated coproducts of twisted primitive elements. The matrix elements are identified with Gasper and Rahman’s multivariate Askey–Wilson polynomials, and from this interpretation we derive their orthogonality relations. Furthermore, the matrix elements are shown to be eigenfunctions of the twisted primitive elements after a change of representation, which gives a quantum algebraic derivation of the fact that the multivariate Askey–Wilson polynomials are solutions of a multivariate bispectral $q$-difference problem.

scholarly journals BCJ, worldsheet quantum algebra and KZ equations

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Chih-Hao Fu ◽  
Yihong Wang
Keyword(s):  
Quantum Group ◽  
Root System ◽  
Open String ◽  
Quantum Algebra ◽  
Twisted Homology

Abstract We exploit the correspondence between twisted homology and quantum group to construct an algebra explanation of the open string kinematic numerator. In this setting the representation depends on string modes, and therefore the cohomology content of the numerator, as well as the location of the punctures. We show that quantum group root system thus identified helps determine the Casimir appears in the Knizhnik-Zamolodchikov connection, which can be used to relate representations associated with different puncture locations.

𝑞-Tricomi functions and quantum algebra representations

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mumtaz Riyasat ◽  
Tabinda Nahid ◽  
Subuhi Khan
Keyword(s):  
Irreducible Representation ◽  
Series Expansion ◽  
Quantum Groups ◽  
Bessel Functions ◽  
Special Functions ◽  
Quantum Algebra ◽  
Two Dimensional ◽  
Quantum Algebras ◽  
Mathematical Software ◽  
First Time

AbstractThe quantum groups nowadays attract a considerable interest of mathematicians and physicists. The theory of 𝑞-special functions has received a group-theoretic interpretation using the techniques of quantum groups and quantum algebras. This paper focuses on introducing the 𝑞-Tricomi functions and 2D 𝑞-Tricomi functions through the generating function and series expansion and for the first time establishing a connecting relation between the 𝑞-Tricomi and 𝑞-Bessel functions. The behavior of these functions is described through shapes, and the contrast between them is observed using mathematical software. Further, the problem of framing the 𝑞-Tricomi and 2D 𝑞-Tricomi functions in the context of the irreducible representation (\omega) of the two-dimensional quantum algebra \mathcal{E}_{q}(2) is addressed, and certain relations involving these functions are obtained. 2-Variable 1-parameter 𝑞-Tricomi functions and their relationship with the 2-variable 1-parameter 𝑞-Bessel functions are also explored.

THE QUANTUM GROUP REALIZATIONS OF 3-D CHERN–SIMONS THEORIES

1995 ◽  
Vol 10 (01) ◽  
pp. 39-49
Author(s):  
C. RAMÍREZ ◽  
L. F. URRUTIA
Keyword(s):  
Quantum Group ◽  
Deformation Parameter ◽  
Canonical Quantization ◽  
Classical Case ◽  
Quantum Algebra ◽  
Simons Theory ◽  
R Matrix ◽  
The One ◽  
Chern Simons ◽  
Chern Simons Theory

The algebra of the integrated connections and of their traces is considered in the one-genus sector of classical and quantum Chern–Simons theory. In the classical case this algebra is braid-like and although the corresponding Jacobi identities are satisfied, the associated r-matrix does not satisfy the classical Yang–Baxter equations. However, it turns out this algebra originates a "quantum" algebra SU (2)q given by its trace algebra. Canonical quantization of the above algebra is performed and a one-parameter expression for the operator ordering is considered. The same quantum algebra with a modified deformation parameter, nontrivially depending on ħ, is obtained.

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