BIREP-Seminar — joint meeting, SBF-projects C3 / A4.

Study group on Borcherds-Kac-Moody algebras

This study group will revolve around Borcherds Lie algebras (or generalised Kac-Moody algebras). The main objectives are to explain the construction of a Littelmann path model for Borcherds algebras, and to give a link between their quantised enveloping algebras and Ringel-Hall algebras. Further related topics (such as crystals and species) may be considered if time permits.

This seminar is a joint meeting with the Stochastics group.

Plan

• In the first talk an overview of the topic will be presented.
• Talks 2 & 3 deal with the definitions and first properties of the objects under consideration. We introduce Borcherds algebras, Weyl groups, real and imaginary roots, integrable representations and characters, trying to focus on a general understanding of the objects (in particular the differences with the usual Kac-Moody case), rather than giving the detailed proofs and constructions.
  Reference: [9, Chapter 2]
• Talks 4 & 5 concern the construction of a Littelmann path model for Borcherds algebras, with an application to a character formula.
  Reference:[6]
• Talk 6 is devoted to the exposition of the following result: The Ringel-Hall algebra of a quiver over a finite field of q elements is the positive part of a quantised enveloping algebra of a Borcherds algebra.
  Reference:[8]

References

1. R.E. Borcherds, Generalized Kac-Moody algebras, J. Algebra 115 (1998), 501–512.
2. R.E. Borcherds, Monstrous moonshine and monstrous Lie superalgebras, Invent. Math. 109 (1992), 405–444.
3. B. Deng, J. Xiao, A new approach to Kac's theorem on representations of valued quivers, Math. Z. 245 (2003), 183–199.
4. K. Jeong, S.-J. Kang, M. Kashiwara, Crystal bases for quantum generalized Kac-Moody algebras, Proc. London Math. Soc. (3) 90 (2005), 395–438.
5. K. Jeong, S.-J. Kang, M. Kashiwara, D.-U. Shin, Abstract crystals for quantum generalised Kac-Moody algebras, Int. Math. Res. Not. IMRN, no 1, Art. ID mm001.
6. A. Joseph, P. Lamprou, A Littelmann path model for crystals of generalized Kac-Moody algebras, arXiv:0804.1485.
7. S.-J. Kang, Quantum deformations of generalized Kac-Moody algebras and their modules, J. Algebra 175 (1995), 1041–1066.
8. B. Sevenhant, M. van den Bergh, A relation between a conjecture of Kac and the structure of the Hall algebra, J. Pure and Appl. Algebra 160 (2001), 319–332.
9. M. Wakimoto, Infinite-dimensional Lie algebras, Translations of Mathematical Monographs, vol. 195, 2001.

Supporting documents

The slides of some talks are now available for downloading:

• The slides of Jean-Marie's talk.
• The slides of Angela's talk.
• The slides of Xenia's talk (the file contains both part I and II).

Prerequisites

While we will try to limit the formal prerequisites to a minimal amount, it could be good to be somewhat familiar with the theory of semi-simple Lie algebras (otherwise, just keep the example of sln in mind). Having followed these lectures might be a plus, but it is by no means essential.