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].

scholarly journals UNIVERSAL CALABI–YAU ALGEBRA: TOWARDS AN UNIFICATION OF COMPLEX GEOMETRY

2003 ◽  
Vol 18 (30) ◽  
pp. 5541-5612 ◽  
Author(s):  
F. ANSELMO ◽  
J. ELLIS ◽  
D. V. NANOPOULOS ◽  
G. VOLKOV
Keyword(s):  
Lie Algebras ◽  
Complex Geometry ◽  
Algebraic Approach ◽  
Higher Dimensions ◽  
Normal Algebra ◽  
Natural Extensions ◽  
Recurrence Formulae ◽  
Different Dimensions ◽  
Arbitrary Dimensions

We present a universal normal algebra suitable for constructing and classifying Calabi–Yau spaces in arbitrary dimensions. This algebraic approach includes natural extensions of reflexive weight vectors to higher dimensions, related to Batyrev's reflexive polyhedra, and their n-ary combinations. It also includes a "dual" construction based on the Diophantine decomposition of invariant monomials, which provides explicit recurrence formulae for the numbers of Calabi–Yau spaces in arbitrary dimensions with Weierstrass, K3, etc., fibrations. Our approach also yields simple algebraic relations between chains of Calabi–Yau spaces in different dimensions, and concrete visualizations of their singularities related to Cartan–Lie algebras. This Universal Calabi–Yau algebra is a powerful tool for deciphering the Calabi–Yau genome in all dimensions.

Extending the Search for Symmetries in the Genetic Code

2003 ◽  
Vol 17 (17) ◽  
pp. 3135-3204 ◽  
Author(s):  
Fernando Antoneli ◽  
Lígia Braggion ◽  
Michael Forger ◽  
José Eduardo M. Hornos
Keyword(s):  
Genetic Code ◽  
Lie Algebras ◽  
Algebraic Approach ◽  
Simple Lie Algebras ◽  
Symmetry Breakings

We report on the search for symmetries in the genetic code involving the medium rank simple Lie algebras [Formula: see text] and [Formula: see text], in the context of the algebraic approach originally proposed by one of the present authors, which aims at explaining the degeneracies encountered in the genetic code as the result of a sequence of symmetry breakings that have occurred during its evolution.

LIE-ALGEBRAIC APPROACH TO THE PROBLEM OF QUASI-EXACT SOLUBILITY IN QUANTUM MECHANICS

1990 ◽  
Vol 05 (23) ◽  
pp. 1891-1899 ◽  
Author(s):  
A. G. USHVERIDZE
Keyword(s):  
Quantum Mechanics ◽  
Lie Algebras ◽  
Algebraic Approach ◽  
New Method ◽  
Exactly Solvable Models ◽  
Solvable Models ◽  
Simple Lie Algebras ◽  
Infinite Dimensional ◽  
Exactly Solvable

A new method of constructing quasi-exactly solvable models of quantum mechanics is proposed. This method is based on the use of infinite-dimensional representations of simple and semi-simple Lie algebras.

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.

scholarly journals Algebraic approach to Rump’s results on relations between braces and pre-lie algebras

2020 ◽  
pp. 2250054
Author(s):  
Agata Smoktunowicz
Keyword(s):  
Lie Algebras ◽  
Characteristic Zero ◽  
Algebraic Approach ◽  
Geometric Approach ◽  
Prime Characteristic ◽  
Affine Manifolds ◽  
Algebraic Interpretation

In 2014, Wolfgang Rump showed that there exists a correspondence between left nilpotent right [Formula: see text]-braces and pre-Lie algebras. This correspondence, established using a geometric approach related to flat affine manifolds and affine torsors, works locally. In this paper, we explain Rump’s correspondence using only algebraic formulae. An algebraic interpretation of the correspondence works for fields of sufficiently large prime characteristic as well as for fields of characteristic zero.

scholarly journals Quantum counterparts of three-dimensional real Lie algebras over harmonic oscillator

Open Physics ◽  
2010 ◽  
Vol 8 (3) ◽  
Author(s):  
Eugen Paal ◽  
Jüri Virkepu
Keyword(s):  
Harmonic Oscillator ◽  
Lie Algebras ◽  
Three Dimensional ◽  
Quantum Algebras ◽  
Jacobi Operators

AbstractOperadic Lax representations for the harmonic oscillator are used to construct the quantum counterparts of three-dimensional (3D) real Lie algebras in Bianchi classification. The Jacobi operators of the quantum algebras are found.

A New Class of Functions Suggested by the Generalized Basic Hypergeometric Function

2017 ◽  
Vol 84 (3-4) ◽  
pp. 161
Author(s):  
Meera H. Chudasama ◽  
B. I. Dave
Keyword(s):  
Hypergeometric Function ◽  
Infinite Order ◽  
Special Functions ◽  
Difference Operators ◽  
Order Zero ◽  
New Class ◽  
Basic Hypergeometric Function ◽  
Order Difference Equation ◽  
The Difference ◽  
Summation Index

We introduce an extended generalized basic hypergeometric function rΦs+p in which p tends to infinity together with the summation index. We define the difference operators and obtain infinite order difference equation, for which these new special functions are eigen functions. We derive some properties, as the order zero of this function, differential equation involving a particular hyper-Bessel type operators of infinite order, and contiguous function relations. A transformation formula and an l-analogue of the q-Maclaurin's series are also obtained.

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.

Sign in / Sign up

Export Citation Format

Share Document