Contents ¶

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

Plan ¶

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(SL_q(2)) [Man88, §1], [Kas95, §IV]

The Hopf algebra U_q(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 U_q(sl(2)) [Kas95, §VI.1-VI.2, VII.1], dual pairing with O_q(SL(2)) [Kas95, §VII.4]
• representations of U(sl(2)) and of U_q(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 U_q(sl(2)) [Kas95, §XI.6-XI.7]

Literature ¶

Bax89 ¶

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

DFMS97 ¶

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

ID89

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.

Man88 ¶

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