Seminar “Yang-Baxter operators”

WS 2013/14


The aim of this seminar is to understand the role of Yang-Baxter operators in and the connections between the following areas of physics and mathematics:

  • exactly solvable models in statistical mechanics
  • invariants of knots and links
  • Hopf algebras and quantum groups


The following plan is also available as a PDF file.

Yang-Baxter operators (Huichi Huang)

[KRT97, §1]

  • (tensor products of vector spaces, linear maps and algebras) [Kas95, §II.1]
  • definition and examples of R-matrices (=Yang-Baxter operators)
  • Artin's definition of the braid group
  • presentation of the braid group in terms of generators and relations
  • representations of the braid group arising from Yang-Baxter operators

Statistische Physik und Ising-Modell (Jins de Jong)

  • Grundlagen (Boltzmann-Verteilung, Zustandssumme, thermodynamische Größen) [Bax89, §1.4], [DFMS97, §3.1]
  • Spin-Systeme, kritische Phänomene [Bax89, §1.1], [DFMS97, § 3.2]
  • 1-dimensionales Ising-Modell, Transfer-Matrix [Bax89, § 2.1]
  • Ausblick: 2-dimensionales Ising-Modell [ID89, § 6.4]

2-dimensionales Ising-Modell (Carlos Pérez)

  • Stern-Dreieck-Dualität [Bax89, §6.4]
  • Eigenschaften der Transfer-Matrizen [Bax89, §7.2-7.5], Eigenwerte [Bax89, §7.6-7.7]
  • Kritisches Verhalten [Bax89, §7.12],

6- und 8-Vertexmodell (Raimar Wulkenhaar)

Notes for the talk

  • Überblick Eis-Modell [Bax89, §8.1]
  • Kommutierende Transfer-Matrizen und Eigenwerte [Bax89, § 9.2-9.6]
  • 8-Vertexmodell [Bax89, §10.1+10.4], [Jim89, pp. 14-49], [Jim89, pp. 50-68]

The Jones Polynomial (Ulrich Pennig)

[KRT97, §5.1-5.2]

  • links, framed links and isotopy
  • link diagrams and Reidemeister moves
  • skein classes, bracket polynomial and Jones polynomial

The category of tangles (Matthias Kemper)

  • tangles, tangle diagrams and Reidemeister moves [KRT97, § 5.4] and [Kas95, X.5]
  • the language of categories and functors [Kas95, §XI.1]
  • strict monoidal categories, tensor functors and the braid category [KRT97, §2.2]; see also [Kas95, §XI.2]
  • braided monoidal categories [KRT97, § 2.3]; see also [Kas95, §XIII.1-2]
  • the braided monoidal categories of tangles and of tangle diagrams [Kas95, §XII.2-3]
  • representations of the tangle category and enhanced R-matrices [Kas95, §XII.4]

Hopf algebras (Michael Holl)

[KRT97, §2.1]

  • definition of bialgebras and Hopf algebras
  • examples, e.g. tensor algebra, group algebra, co-opposite, dual
  • properties of the antipode
  • example O(SLq(2)) [Man88, §1], [Kas95, §IV]

The Hopf algebra Uq(sl(2)) (Martijn Caspers)

  • the Lie algebra sl(2) [Kas95, §5.3]
  • the enveloping algebra of a Lie algebra and its Hopf algebra structure [Kas95, §5.2]
  • dual pairings of Hopf algebras and the pairing between U(sl(2)) and O(SL(2)) [Kas95, §5.7]
  • q-calculus and definition of Uq(sl(2)) [Kas95, §VI.1-VI.2, VII.1], dual pairing with Oq(SL(2)) [Kas95, §VII.4]
  • representations of U(sl(2)) and of Uq(sl(2)) [Kas95, §V.4, VI.3, VI.5]

Braided Hopf algebras and the Drinfeld double (Jianchao Wu)

  • braided Hopf algebras and R-matrices [KRT97, §2.4]
  • the Drinfeld double construction [KRT97, §3]
  • application to Uq(sl(2)) [Kas95, §XI.6-XI.7]



Rodney J. Baxter, Exactly solved models in statistical mechanics, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, 1989, Reprint of the 1982 original.


Philippe Di Francesco, Pierre Mathieu, and David Sénéchal, Conformal field theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York, 1997.


Claude Itzykson and Jean-Michel Drouffe, Statistical field theory. Vol. 1, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1989, From Brownian motion to renormalization and lattice gauge theory. ### Jim89 Michio Jimbo (ed.), Yang-Baxter equation in integrable systems, Advanced Series in Mathematical Physics, vol. 10, World Scientific Publishing Co. Inc., Teaneck, NJ, 1989. ### Kas95 Christian Kassel, Quantum groups, Graduate Texts in Mathematics, vol. 155, Springer-Verlag, New York, 1995. ### KRT97 Christian Kassel, Marc Rosso, and Vladimir Turaev, Quantum groups and knot invariants, Panoramas et Synthèses [Panoramas and Syntheses], vol. 5, Société Mathématique de France, Paris, 1997.


Yu. I. Manin, Quantum groups and noncommutative geometry, Université de Montréal Centre de Recherches Mathématiques, Montreal, QC, 1988.