Boson realizations of quantum algebras and some physical spaces with quantum algebra symmetry

1993 ◽  
Vol 27 (4) ◽  
pp. 273-277
Author(s):  
Mao-Zheng Guo ◽  
Xu-Feng Liu ◽  
Min Qian
Keyword(s):  
Quantum Algebra ◽  
Quantum Algebras ◽  
Physical Spaces

Quantum algebra embeddings: deforming functionals and algebraic approach

1994 ◽  
Vol 72 (7-8) ◽  
pp. 519-526 ◽  
Author(s):  
J. Van der Jeugt
Keyword(s):  
Hypergeometric Function ◽  
Lie Algebras ◽  
Algebraic Approach ◽  
Quantum Algebra ◽  
Physical Models ◽  
Quantum Algebras ◽  
Basic Hypergeometric Function

The study of subalgebras of Lie algebras arising in physical models has been important for many applications. In the present paper we examine the q-deformation of such embeddings; the Lie algebras are then replaced by quantum algebras. Two methods are presented: one based upon deforming functionals, and a direct algebraic approach. A number of examples are given, e.g., [Formula: see text] and [Formula: see text]. For the last example, we give the q-boson construction, and the relevant overlap coefficients are related to a generalized basic hypergeometric function [Formula: see text].

APPLICATIONS OF COMPUTER ALGEBRA IN QUANTUM GROUPS

1994 ◽  
Vol 05 (04) ◽  
pp. 701-706
Author(s):  
W.-H. STEEB
Keyword(s):  
Quantum Groups ◽  
Computer Algebra ◽  
Kronecker Product ◽  
Quantum Algebra ◽  
Theoretical Physics ◽  
Baxter Equation ◽  
Quantum Algebras ◽  
Helpful Tool

Quantum groups and quantum algebras play a central role in theoretical physics. We show that computer algebra is a helpful tool in the investigations of quantum groups. We give an implementation of the Kronecker product together with the Yang-Baxter equation. Furthermore the quantum algebra obtained from the Yang-Baxter equation is implemented. We apply the computer algebra package REDUCE.

Canonical bases of modified quantum algebras for type A2

2018 ◽  
Vol 17 (06) ◽  
pp. 1850113
Author(s):  
Weideng Cui
Keyword(s):  
Canonical Basis ◽  
Quantum Algebra ◽  
Explicit Description ◽  
Quantum Algebras ◽  
Canonical Bases ◽  
Type Formula

The modified quantum algebra [Formula: see text] associated to a quantum algebra [Formula: see text] was introduced by Lusztig. [Formula: see text] has a remarkable basis, which was defined by Lusztig, called the canonical basis. In this paper, we give an explicit description of all elements of the canonical basis of [Formula: see text] for type [Formula: see text].

SYMMETRY, INTEGRABILITY AND DEFORMATIONS OF LONG-RANGE INTERACTING HAMILTONIANS

1999 ◽  
Vol 13 (24n25) ◽  
pp. 2903-2908 ◽  
Author(s):  
ANGEL BALLESTEROS
Keyword(s):  
Long Range ◽  
Hamiltonian Systems ◽  
Quantum Algebra ◽  
Complete Integrability ◽  
Quantum Algebras ◽  
R Matrix ◽  
Completely Integrable ◽  
Integrable Deformation ◽  
Coalgebra Structure ◽  
Integrable Deformations

The notion of coalgebra symmetry in Hamiltonian systems is analysed. It is shown how the complete integrability of some long-range interacting Hamiltonians can be extracted from their associated coalgebra structure with no use of a quantum R-matrix. Within this framework, integrable deformations can be considered as direct consequences of the introduction of coalgebra deformations (quantum algebras). As an example, the Gaudin magnet is derived from a sl(2) coalgebra, and a completely integrable deformation of this Hamiltonian is obtained through a twisted gl(2) quantum algebra.

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.

scholarly journals ORIENTED QUANTUM ALGEBRAS, CATEGORIES AND INVARIANTS OF KNOTS AND LINKS

2001 ◽  
Vol 10 (07) ◽  
pp. 1047-1084 ◽  
Author(s):  
LOUIS KAUFFMAN ◽  
DAVID E. RADFORD
Keyword(s):  
Quantum Algebra ◽  
Quantum Algebras ◽  
Link Invariants ◽  
Quantum Link ◽  
Knots And Links

This paper defines the concept of an oriented quantum algebra and develops its application to the construction of quantum link invariants. We show, in fact, that all known quantum link invariants can be put into this framework.

scholarly journals Symmetry and reversibility properties for quantum algebras and skew Poincaré-Birkhoff-Witt extensions

2018 ◽  
Vol 14 (27) ◽  
pp. 29-52 ◽  
Author(s):  
Armando Reyes ◽  
Julio Jaramillo
Keyword(s):  
Quantum Algebra ◽  
Theoretical Physics ◽  
Quantum Algebras ◽  
Coeffi Cients ◽  
Skew Pbw Extensions

Our aim in this paper is to investigate symmetry and reversibility pro-perties for quantum algebras and skew PBW extensions. Under certainconditions we prove that these properties transfer from a ring of coeffi-cients to a quantum algebra or skew PBW extension over this ring. In thisway we generalize several results established in the literature and consideralgebras which have not been studied before. We illustrate our results withremarkable examples of theoretical physics

KILLING FORMS, HARISH-CHANDRA ISOMORPHISMS, AND UNIVERSAL R-MATRICES FOR QUANTUM ALGEBRAS

1992 ◽  
Vol 07 (supp01b) ◽  
pp. 941-961 ◽  
Author(s):  
TOSHIYUKI TANISAKI
Keyword(s):  
Structure Theorem ◽  
Quantum Algebra ◽  
Quantum Algebras ◽  
Killing Form ◽  
R Matrix ◽  
Detailed Proof ◽  
Duality Pairing

We describe the Killing form of the quantum algebra using the duality pairing between the plus and the minus parts, and give a structure theorem for the center. A detailed proof of the existence of the universal R-matrix (Drinfeld's theorem) is also given.

scholarly journals The tensor product of two oriented quantum algebras

2019 ◽  
Vol 19 (06) ◽  
pp. 2050104
Author(s):  
Tianshui Ma ◽  
Haiyan Yang ◽  
Tao Yang
Keyword(s):  
Tensor Product ◽  
Knot Theory ◽  
Quantum Algebra ◽  
Quantum Algebras ◽  
Special Case

In this paper, we give the oriented quantum algebra (OQA) structures on the tensor product of two different OQAs by using Chen’s weak [Formula: see text]-matrix in [J. Algebra 204 (1998) 504–531]. As a special case, the OQA structures on the tensor product of an OQA with itself are provided, which are different from Radford’s results in [J. Knot Theory Ramifications 16 (2007) 929–957].

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