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Axioms
Special Issues
Hopf Algebras, Quantum Groups and Yang-Baxter Equations 2014
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Hopf Algebras, Quantum Groups and Yang-Baxter Equations 2014

A special issue of Axioms (ISSN 2075-1680).

Deadline for manuscript submissions: closed (31 December 2014) | Viewed by 18810

Special Issue Editor

Dr. Florin Felix Nichita

E-Mail Website
Guest Editor
Simion Stoilow Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania
Interests: (co)algebras; bialgebras; Yang–Baxter equations; Lie (co)algebras; quantum groups; Hopf algebras; duality theories
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The Yang-Baxter equation first appeared in theoretical physics, in a paper (1968) by the Nobel laureate C.N. Yang, and in statistical mechanics, in R.J. Baxter's work (1971). Later, it turned out that this equation plays a crucial role in: quantum groups, knot theory, braided categories, analysis of integrable systems, quantum mechanics, non-commutative descent theory, quantum computing, non-commutative geometry, etc.

Many scientists have used the axioms of various algebraic structures (quasi-triangular Hopf algebras, Yetter-Drinfeld categories, Lie (super)algebras, algebra structures, Boolean algebras, etc) or computer calculations in order to produce solutions for the Yang-Baxter equation. However, the full classification of its solutions remains an open problem.

Contributions related to the various aspects of the Yang-Baxter equations, the related algebraic structures, and their applications are invited. We would like to gather together both relevant reviews (with historical notes, open problems or research directions) and research papers (on the new developments of the Yang-Baxter equations).

Dr. Florin Felix Nichita
Guest Editor

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Keywords

  • Yang-Baxter equation
  • Yang-Baxter system
  • Quantum Groups
  • Hopf algebra
  • braid group
  • braided category

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Related Special Issue

Published Papers (4 papers)

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Research

241 KiB  
Article
Generalized Yang–Baxter Operators for Dieudonné Modules
by Rui Miguel Saramago
Axioms 2015, 4(2), 177-193; https://doi.org/10.3390/axioms4020177 - 8 May 2015
Cited by 1 | Viewed by 4015
Abstract
An enrichment of a category of Dieudonné modules is made by considering Yang–Baxter conditions, and these are used to obtain ring and coring operations on the corresponding Hopf algebras. Some examples of these induced structures are discussed, including those relating to the Morava [...] Read more.
An enrichment of a category of Dieudonné modules is made by considering Yang–Baxter conditions, and these are used to obtain ring and coring operations on the corresponding Hopf algebras. Some examples of these induced structures are discussed, including those relating to the Morava K-theory of Eilenberg–MacLane spaces. Full article
(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations 2014)
259 KiB  
Article
Convergence Aspects for Generalizations of q-Hypergeometric Functions
by Thomas Ernst
Axioms 2015, 4(2), 134-155; https://doi.org/10.3390/axioms4020134 - 8 Apr 2015
Cited by 1 | Viewed by 4021
Abstract
In an earlier paper, we found transformation and summation formulas for 43 q-hypergeometric functions of 2n variables. The aim of the present article is to find convergence regions and a few conjectures of convergence regions for these functions based on a vector [...] Read more.
In an earlier paper, we found transformation and summation formulas for 43 q-hypergeometric functions of 2n variables. The aim of the present article is to find convergence regions and a few conjectures of convergence regions for these functions based on a vector version of the Nova q-addition. These convergence regions are given in a purely formal way, extending the results of Karlsson (1976). The Γq-function and the q-binomial coefficients, which are used in the proofs, are adjusted accordingly. Furthermore, limits and special cases for the new functions, e.g., q-Lauricella functions and q-Horn functions, are pointed out. Full article
(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations 2014)
435 KiB  
Article
Azumaya Monads and Comonads
by Bachuki Mesablishvili and Robert Wisbauer
Axioms 2015, 4(1), 32-70; https://doi.org/10.3390/axioms4010032 - 19 Jan 2015
Viewed by 4306
Abstract
The definition of Azumaya algebras over commutative rings \(R\) requires the tensor product of modules over \(R\) and the twist map for the tensor product of any two \(R\)-modules. Similar constructions are available in braided monoidal categories, and Azumaya algebras were defined in [...] Read more.
The definition of Azumaya algebras over commutative rings \(R\) requires the tensor product of modules over \(R\) and the twist map for the tensor product of any two \(R\)-modules. Similar constructions are available in braided monoidal categories, and Azumaya algebras were defined in these settings. Here, we introduce Azumaya monads on any category \(\mathbb{A}\) by considering a monad \((F,m,e)\) on \(\mathbb{A}\) endowed with a distributive law \(\lambda: FF\to FF\) satisfying the Yang–Baxter equation (BD%please define -law). This allows to introduce an opposite monad \((F^\lambda,m\cdot \lambda,e)\) and a monad structure on \(FF^\lambda\). The quadruple \((F,m,e,\lambda)\) is called an Azumaya monad, provided that the canonical comparison functor induces an equivalence between the category \(\mathbb{A}\) and the category of \(FF^\lambda\)-modules. Properties and characterizations of these monads are studied, in particular for the case when \(F\) allows for a right adjoint functor. Dual to Azumaya monads, we define Azumaya comonads and investigate the interplay between these notions. In braided categories (V\(,\otimes,I,\tau)\), for any V-algebra \(A\), the braiding induces a BD-law \(\tau_{A,A}:A\otimes A\to A\otimes A\), and \(A\) is called left (right) Azumaya, provided the monad \(A\otimes-\) (resp. \(-\otimes A\)) is Azumaya. If \(\tau\) is a symmetry or if the category V admits equalizers and coequalizers, the notions of left and right Azumaya algebras coincide. Full article
(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations 2014)
216 KiB  
Communication
The Yang-Baxter Equation, (Quantum) Computers and Unifying Theories
by Radu Iordanescu, Florin F. Nichita and Ion M. Nichita
Axioms 2014, 3(4), 360-368; https://doi.org/10.3390/axioms3040360 - 14 Nov 2014
Cited by 10 | Viewed by 5780
Abstract
Quantum mechanics has had an important influence on building computers;nowadays, quantum mechanics principles are used for the processing and transmission ofinformation. The Yang-Baxter equation is related to the universal gates from quantumcomputing and it realizes a unification of certain non-associative structures. Unifyingstructures could [...] Read more.
Quantum mechanics has had an important influence on building computers;nowadays, quantum mechanics principles are used for the processing and transmission ofinformation. The Yang-Baxter equation is related to the universal gates from quantumcomputing and it realizes a unification of certain non-associative structures. Unifyingstructures could be seen as structures which comprise the information contained in other(algebraic) structures. Recently, we gave the axioms of a structure which unifies associativealgebras, Lie algebras and Jordan algebras. Our paper is a review and a continuation of thatapproach. It also contains several geometric considerations. Full article
(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations 2014)
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