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Journals
Axioms
Special Issues
Hopf Algebras, Quantum Groups and Yang–Baxter Equations 2017
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Hopf Algebras, Quantum Groups and Yang–Baxter Equations 2017

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

Deadline for manuscript submissions: closed (30 October 2017) | Viewed by 22628

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 of the Nobel laureate C.N. Yang, and in statistical mechanics, in R.J. Baxter's work. 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 (quasitriangular Hopf algebras, Yetter–Drinfeld categories, Lie (super)algebras, algebra structures, Boolean algebras, brace structures, relations on sets, 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 Equation, the related algebraic structures, and their applications are invited. We would like to gather together relevant reviews, research articles, and communications.

Dr. Florin Felix Nichita
Guest Editor

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Axioms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Yang–Baxter equation
  • quantum groups
  • link invariants
  • virtual knot theory
  • set-theoretical Yang–Baxter equation
  • brace structure
  • quasitriangular Hopf algebra
  • braid group
  • braided category
  • classical Yang–Baxter equation
  • Myhill–Nerode monoid
  • Yang–Baxter system

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Published Papers (5 papers)

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Research

7 pages, 220 KiB  
Article
On Transcendental Numbers: New Results and a Little History
by Solomon Marcus and Florin F. Nichita
Axioms 2018, 7(1), 15; https://doi.org/10.3390/axioms7010015 - 1 Mar 2018
Cited by 10 | Viewed by 5453
Abstract
Bringing toghether mathematical and philosophical ideas related to transcendental numbers, this paper begins with a survey on transcendence and transcendental numbers, it then presents several properties of the transcendental numbers e and π , and then it gives the proof of a new [...] Read more.
Bringing toghether mathematical and philosophical ideas related to transcendental numbers, this paper begins with a survey on transcendence and transcendental numbers, it then presents several properties of the transcendental numbers e and π , and then it gives the proof of a new inequality for transcendental numbers. Also, in relationship with these topics, we study solutions to the Yang-Baxter equation from hyperbolic functions and from logical implication. Full article
(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang–Baxter Equations 2017)
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13 pages, 215 KiB  
Article
On Solutions to the Set-Theoretical Yang-Baxter Equation in Wajsberg-Algebras
by Tahsin Oner and Tugce Katican
Axioms 2018, 7(1), 6; https://doi.org/10.3390/axioms7010006 - 20 Jan 2018
Cited by 9 | Viewed by 4155
Abstract
In this work, we introduce Wajsberg algebras which are equivalent structures to MV-algebras in their implicational version, and then we define new notions and give new solutions to the set-theoretical Yang-Baxter equation by using Wajsberg algebras. Full article
(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang–Baxter Equations 2017)
275 KiB  
Article
Universal Enveloping Commutative Rota–Baxter Algebras of Pre- and Post-Commutative Algebras
by Vsevolod Gubarev
Axioms 2017, 6(4), 33; https://doi.org/10.3390/axioms6040033 - 7 Dec 2017
Cited by 2 | Viewed by 3434
Abstract
Universal enveloping commutative Rota–Baxter algebras of pre- and post-commutative algebras are constructed. The pair of varieties (RBλCom, postCom) is proved to be a Poincaré–Birkhoff–Witt-pair (PBW)-pair and the pair (RBCom, preCom) is proven not to be. Full article
(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang–Baxter Equations 2017)
506 KiB  
Article
Factorization of Graded Traces on Nichols Algebras
by Simon Lentner and Andreas Lochmann
Axioms 2017, 6(4), 32; https://doi.org/10.3390/axioms6040032 - 4 Dec 2017
Viewed by 3532
Abstract
A ubiquitous observation for finite-dimensional Nichols algebras is that as a graded algebra the Hilbert series factorizes into cyclotomic polynomials. For Nichols algebras of diagonal type (e.g., Borel parts of quantum groups), this is a consequence of the existence of a root system [...] Read more.
A ubiquitous observation for finite-dimensional Nichols algebras is that as a graded algebra the Hilbert series factorizes into cyclotomic polynomials. For Nichols algebras of diagonal type (e.g., Borel parts of quantum groups), this is a consequence of the existence of a root system and a Poincare-Birkhoff-Witt (PBW) basis basis, but, for nondiagonal examples (e.g., Fomin–Kirillov algebras), this is an ongoing surprise. In this article, we discuss this phenomenon and observe that it continues to hold for the graded character of the involved group and for automorphisms. First, we discuss thoroughly the diagonal case. Then, we prove factorization for a large class of nondiagonal Nichols algebras obtained by the folding construction. We conclude empirically by listing all remaining examples, which were in size accessible to the computer algebra system GAP and find that again all graded characters factorize. Full article
(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang–Baxter Equations 2017)
707 KiB  
Article
An Independent Set of Axioms of MV-Algebras and Solutions of the Set-Theoretical Yang–Baxter Equation
by Tahsin Oner, Ibrahim Senturk and Gulsah Oner
Axioms 2017, 6(3), 17; https://doi.org/10.3390/axioms6030017 - 22 Jun 2017
Cited by 17 | Viewed by 5241
Abstract
The aim of this paper is to give a new equivalent set of axioms for MV-algebras, and to show that the axioms are independent. In addition to this, we handle Yang–Baxter equation problem. In conclusion, we construct a new set-theoretical solution for [...] Read more.
The aim of this paper is to give a new equivalent set of axioms for MV-algebras, and to show that the axioms are independent. In addition to this, we handle Yang–Baxter equation problem. In conclusion, we construct a new set-theoretical solution for the Yang–Baxter equation by using MV-algebras. Full article
(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang–Baxter Equations 2017)
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