Special Issue "Hopf Algebras, Quantum Groups and Yang-Baxter Equations 2014"
A special issue of Axioms (ISSN 2075-1680).
Deadline for manuscript submissions: 31 December 2014
Dr. Florin Felix Nichita
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
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
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.
Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are refereed through a 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. For the first couple of issues the Article Processing Charge (APC) will be waived for well-prepared manuscripts. English correction and/or formatting fees of 250 CHF (Swiss Francs) will be charged in certain cases for those articles accepted for publication that require extensive additional formatting and/or English corrections.
- Yang-Baxter equation
- Yang-Baxter system
- Quantum Groups
- Hopf algebra
- braid group
- braided category
The below list represents only planned manuscripts. Some of these manuscripts have not been received by the Editorial Office yet. Papers submitted to MDPI journals are subject to peer-review.
Type of paper: article
Title: Virtual Knot Theory and Quantum Link Invarints
Author: Louis H. Kauffman
Abstract: Virtual knot theory is an extension of classical knot theory that studies knots in thickened orientable surfaces and has a diagrammatic expression using classical knot diagrams with extra crossings called virtual crossings. In the diagrammatic form, virtual knots and links are represented by virtual diagrams up to Reidemeister moves plus Detour moves where a detour move replaces an arc with a consecutive sequence of virtual crossings by another arc whose intersections with the remaining diagram are also virtual. A significant restriction of virtual knot theory is Rotational virtual knot theory where the detour moves are represented by regular homotopies of the replacement arc. We have the Theorem. Every quantum link invariant for classcial links extends to an invariant for rotational virtual links. Quantum link invariants are based on solutions to the Yang-Baxter equation and particluarly on solutions to the Yang-Baxter equation that come from Hopf algebras and quantum groups. This theorem provides a rich field of invariants for studying rotational virtual knots. It shows that the proper domain for quantum link invariants is virtual knot theory. The paper explores this connection in detail, with background on virtual knot theory, background on quantum link invariants, and new results about the interrelationship of these fields.
Type of paper: Article
Title: Quasitriangular structures on monoid rings and their duals
Author name: Nigel Byott
Abstract: We consider quasitriangular structures on the bialgebras obtained as monoid rings of finite monoids, and on the dual bialgebras of these. We pay particular attention to the Myhill-Nerode monoids arising from the algebraic formulation, due to W. Nichols and R. Underwood, of the Myhill-Nerode theorem in theoretical computer science. For one example of such a monoid, Underwood [Axioms 1, (2012), 155-172] showed that the monoid ring and its dual each admit a unique quasitriangular structure. In this paper, we show that the same holds for many other examples of Myhill-Nerode monoids, and that this remains true if the definition of quasitriangular structure is weakened by removing the almost cocommutativity condition.
Last update: 2 September 2014