Special Issue "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)

Special Issue Editor

Guest Editor
Dr. Florin Felix Nichita

Simion Stoilow Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania
Website | E-Mail
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 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

Manuscript Submission Information

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. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short 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.

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 quarterly 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 350 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

Published Papers (3 papers)

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Research

Open AccessArticle Universal Enveloping Commutative Rota–Baxter Algebras of Pre- and Post-Commutative Algebras
Axioms 2017, 6(4), 33; doi:10.3390/axioms6040033 (registering DOI)
Received: 2 October 2017 / Revised: 22 November 2017 / Accepted: 4 December 2017 / Published: 7 December 2017
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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)
Open AccessArticle Factorization of Graded Traces on Nichols Algebras
Axioms 2017, 6(4), 32; doi:10.3390/axioms6040032
Received: 27 October 2017 / Revised: 23 November 2017 / Accepted: 25 November 2017 / Published: 4 December 2017
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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)
Open AccessArticle An Independent Set of Axioms of MV-Algebras and Solutions of the Set-Theoretical Yang–Baxter Equation
Axioms 2017, 6(3), 17; doi:10.3390/axioms6030017
Received: 3 May 2017 / Revised: 18 June 2017 / Accepted: 20 June 2017 / Published: 22 June 2017
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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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