axioms Hopf Algebras, Quantum Groups and Yang-Baxter Equations Edited by Florin Felix Nichita Printed Edition of the Special Issue Published in Axioms www.mdpi.com/journal/axioms Hopf Algebras, Quantum Groups and Yang-Baxter Equations Hopf Algebras, Quantum Groups and Yang-Baxter Equations Special Issue Editor Florin Felix Nichita MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade Special Issue Editor Florin Felix Nichita Simion Stoilow Institute of Mathematics of the Romanian Academy Romania Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Axioms (ISSN 2075-1680) in 2012 (available at: https://www.mdpi.com/journal/axioms/special issues/ hopf algebras) For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year, Article Number, Page Range. ISBN 978-3-03897-324-9 (Pbk) ISBN 978-3-03897-325-6 (PDF) c 2019 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND. Contents About the Special Issue Editor ...................................... vii Preface to ”Hopf Algebras, Quantum Groups and Yang-Baxter Equations” ............ ix Tomasz Brzezinski ´ and Simon A. Fairfax Bundles over Quantum Real Weighted Projective Spaces Reprinted from: Axioms 2012, 1, 201–225, doi:10.3390/axioms1020201 ................ 1 Michiel Hazewinkel Hasse-Schmidt Derivations and the Hopf Algebra of Non-Commutative Symmetric Functions Reprinted from: Axioms 2012, 1, 149–154, doi:10.3390/axioms1020149 ................ 23 Alexander E. Hoffnung The Hecke Bicategory Reprinted from: Axioms 2012, 1, 291–323, doi:10.3390/axioms1030291 ................ 29 Konstantinos Kanakoglou Gradings, Braidings, Representations, Paraparticles: Some Open Problems Reprinted from: Axioms 2012, 1, 74–98, doi:10.3390/axioms1010074 ................. 57 Apoorva Khare The Sum of a Finite Group of Weights of a Hopf Algebra Reprinted from: Axioms 2012, 1, 259–290, doi:10.3390/axioms1030259 ................ 77 Joel Lemay Valued Graphs and the Representation Theory of Lie Algebras Reprinted from: Axioms 2012, 1, 111–148, doi:10.3390/axioms1020111 ................104 Jon Links Hopf Algebra Symmetries of an Integrable Hamiltonian for Anyonic Pairing Reprinted from: Axioms 2012, 1, 226–237, doi:10.3390/axioms1020226 ................137 Florin F. Nichita and Bartosz Zielinski ´ The Duality between Corings and Ring Extensions Reprinted from: Axioms 2012, 1, 74–185, doi:10.3390/axioms1020173 .................148 Jeffrey R. Schmidt From Coalgebra to Bialgebra for the Six-Vertex Model: The Star-Triangle Relation as a Necessary Condition for Commuting Transfer Matrices Reprinted from: Axioms 2012, 1, 186–200, doi:10.3390/axioms1020186 ................160 Kenichi Shimizu Frobenius–Schur Indicator for Categories with Duality Reprinted from: Axioms 2012, 1, 324–364, doi:10.3390/axioms1030324 ................174 Robert G. Underwood Quasitriangular Structure of Myhill–Nerode Bialgebras Reprinted from: Axioms 2012, 1, 155–172, doi:10.3390/axioms1020155 ................210 v About the Special Issue Editor Florin Felix Nichita, Senior Researcher, Ph.D. (SUNY Buffalo, 2001), currently works in the Topology and Differential Geometry Department at the Simion Stoilow Institute of Mathematics of the Romanian Academy (IMAR). He is a member of the Editorial Board of Axioms, a member of the Advisory Board of Sci, and has acted as a Guest Editor for Axioms since 2012. He was a Marie Curie Research Fellow in the Mathematics Department, Swansea University, Wales, UK (2003–2005). He has organized International Workshops on Differential Geometry and its Applications (Ploiesti, Romania). He has taught, given talks, or made presentations in Boston, University of Notre Dame, Cambridge, Atlanta, Gregynog, New Paltz (NY), Malta, Praga, Barcelona, Sofia, Kuwait City, Iasi, Brasov, Alba Iulia, and Targu Mures, among others. One of his most recent publication in Axioms is entitled “On Transcendental Numbers: New Results and a Little History” (with Solomon Marcus). vii Preface to ”Hopf Algebras, Quantum Groups and Yang-Baxter Equations” Various aspects of the Yang-Baxter equation, related algebraic structures, and applications are presented in this volume. The algebraic approach to bundles in non-commutative geometry and the definition of quantum real weighted projective spaces are reviewed in ”Bundles over Quantum Real Weighted Projective Spaces”, by Tomasz Brzezinski´ and Simon A. Fairfax. Let NSymm be the Hopf algebra of non-commutative symmetric functions. In ”Hasse-Schmidt Derivations and the Hopf Algebra of Non-Commutative Symmetric Functions”, by Michiel Hazewinkel, it is shown that an associative algebra A with a Hasse-Schmidt derivation on it is exactly the same as an NSymm module algebra. An application of the program of groupoidification, leading up to a sketch of a categorification of the Hecke algebroid, is presented in the article ”The Hecke Bicategory” by Alexander E. Hoffnung. ”Gradings, Braidings, Representations, Paraparticles: Some Open Problems”, by Konstantinos Kanakoglou, is a research proposal on the algebraic structure, the representations, and the possible applications of paraparticle algebras. Apoorva Khare, in the article ”The Sum of a Finite Group of Weights of a Hopf Algebra”, evaluates the sum of a finite group of linear characters of a Hopf algebra, at all grouplike and skew-primitive elements. ”Valued Graphs and the Representation Theory of Lie Algebras”, by Joel Lemay, deals with quivers (directed graphs), species (a generalization of quivers), and their representations (which play a key role in many areas of mathematics including combinatorics, geometry, and algebra). ”Hopf Algebra Symmetries of an Integrable Hamiltonian for Anyonic Pairing”, by Jon Links, is a mathematical physics paper. ”The Duality between Corings and Ring Extensions”, by Florin F. Nichita and Bartosz Zielinski, deals with an extension for the duality between corings and ring extensions. ”From Coalgebra to Bialgebra for the Six-Vertex Model: The Star-Triangle Relation as a Necessary Condition for Commuting Transfer Matrices”, by Jeffrey R. Schmidt, is a mathematical physics paper. ”Frobenius-Schur Indicator for Categories with Duality”, by Kenichi Shimizu, introduces the Frobenius–Schur indicator for categories with duality. ”Quasitriangular Structure of Myhill–Nerode Bialgebras”, by Robert G. Underwood, investigates the quasitriangular structure of Myhill-Nerode bialgebras. I would like to thank the authors who contributed to this volume, in addition to the referees and the editorial staff of Axioms. Florin Felix Nichita Special Issue Editor ix axioms Article Bundles over Quantum Real Weighted Projective Spaces Tomasz Brzezinski ´ * and Simon A. Fairfax Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, UK; E-Mail: [email protected] * E-Mail: [email protected]; Tel.: +44-1792-295460; Fax: +44-1792-295843. Received: 10 July 2012; in revised form: 21 August 2012 / Accepted: 23 August 2012 / Published: 17 September 2012 Abstract: The algebraic approach to bundles in non-commutative geometry and the Definition of quantum real weighted projective spaces are reviewed. Principal U(1)-bundles over quantum real weighted projective spaces are constructed. As the spaces in question fall into two separate classes, the negative or odd class that generalises quantum real projective planes and the positive or even class that generalises the quantum disc, so do the constructed principal bundles. In the negative case the principal bundle is proven to be non-trivial and associated projective modules are described. In the positive case the principal bundles turn out to be trivial, and so all the associated modules are free. It is also shown that the circle (co)actions on the quantum Seifert manifold that define quantum real weighted projective spaces are almost free. Keywords: quantum real weighted projective space; principal comodule algebra; noncommutative line bundle 1. Introduction In an algebraic setup an action of a circle on a quantum space corresponds to a coaction of a Hopf algebra of Laurent polynomials in one variable on the noncommutative coordinate algebra of the quantum space. Such a coaction can equivalently be understood as a Z-grading of this coordinate algebra. A typical Z-grading assigns degree ±1 to every generator of this algebra (different from the identity). The degree zero part forms a subalgebra which in particular cases corresponds to quantum complex or real projective spaces (grading of coordinate algebras of quantum spheres [1]or prolonged quantum spheres [2]). Often this grading is strong, meaning that the product of i, j-graded parts is equal to the i + j-part of the total algebra. In geometric terms this reflects the freeness of the circle action. In two recent papers [3,4] circle actions on three-dimensional (and, briefly, higher dimensional) quantum spaces were revisited. Rather than assigning a uniform grade to each generator, separate