Special Issue "Hopf Algebras, Quantum Groups and Yang-Baxter Equations 2014"

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

Deadline for manuscript submissions: closed (31 December 2014)

Special Issue Editor

Guest Editor
Dr. Florin Felix Nichita (Website)

Simion Stoilow Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania
Phone: + 40 244 598 194
Interests: (co)algebras; bialgebras; Yang-Baxter equations; Lie (co)algebras; quantum groups; Hopf algebras; duality theories

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

Submission

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. Papers will be published continuously (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

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Keywords

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

Related Special Issue

Published Papers (4 papers)

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Research

Open AccessArticle Generalized Yang–Baxter Operators for Dieudonné Modules
Axioms 2015, 4(2), 177-193; doi:10.3390/axioms4020177
Received: 19 December 2014 / Accepted: 30 April 2015 / Published: 8 May 2015
Cited by 1 | PDF Full-text (241 KB) | HTML Full-text | XML Full-text
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 [...] 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)
Open AccessArticle Convergence Aspects for Generalizations of q-Hypergeometric Functions
Axioms 2015, 4(2), 134-155; doi:10.3390/axioms4020134
Received: 6 September 2014 / Revised: 8 March 2015 / Accepted: 31 March 2015 / Published: 8 April 2015
PDF Full-text (259 KB) | HTML Full-text | XML Full-text
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 [...] 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)
Open AccessArticle Azumaya Monads and Comonads
Axioms 2015, 4(1), 32-70; doi:10.3390/axioms4010032
Received: 1 October 2014 / Revised: 11 December 2014 / Accepted: 4 January 2015 / Published: 19 January 2015
PDF Full-text (435 KB) | HTML Full-text | XML Full-text
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 [...] 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)
Open AccessCommunication The Yang-Baxter Equation, (Quantum) Computers and Unifying Theories
Axioms 2014, 3(4), 360-368; doi:10.3390/axioms3040360
Received: 22 September 2014 / Revised: 25 October 2014 / Accepted: 4 November 2014 / Published: 14 November 2014
Cited by 1 | PDF Full-text (216 KB) | HTML Full-text | XML Full-text
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 [...] 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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