Abstract
This paper outlines an algorithm for computing structure constants of Kac–Moody Lie algebras. In contrast to the methods currently used for finite-dimensional Lie algebras, which rely on the additive structure of the roots, it reduces to computations in the extended Weyl group first defined by Jacques Tits in about 1966. The new algorithm has some theoretical interest, and its basis is a mathematical result generalizing a theorem of Tits about the finite-dimensional case. The explicit algorithm seems to be new, however, even in the finite-dimensional case. I include towards the end some remarks about repetitive patterns of structure constants, which I expect to play an important role in understanding the associated groups. That neither the idea of Tits nor the phenomenon of repetition has already been exploited I take as an indication of how little we know about Kac–Moody structures.
To Nolan Wallach, wishing him many more years of achievement
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References
Brigitte Brink and Robert Howlett, A finiteness property and an automatic structure for Coxeter groups, Mathematische Annalen 296 (1993), 179–190.
Bill Casselman, Computations in Coxeter groups II. Constructing minimal roots, Representation Theory 12 (2008), 260–293.
Roger Carter, Simple groups of Lie type, Wiley, 1972.
Claude Chevalley, Sur certains groupes simples, Tohoku Mathematics Journal 48 (1955), 14–66.
Arjeh Cohen, Scott Murray, and Don Taylor, Computing in groups of Lie type, Mathematics of Computation 73 (2004), 1477–1498.
Igor Frenkel and Victor Kac, Affine Lie algebras and dual resonance models, Inventiones Mathematicae 62 (1980), 23–66.
Victor Kac, Infinite Dimensional Lie Algebras (second edition). Cambridge University Press, Cambridge, New York, 1985.
V. G. Kac et D. H. Peterson, ‘Defining relations of certain infinite dimensional groups’, in Élie Cartan et les mathématiques d’aujourd’hui, Astérisque, volume hors série (1985), 165–208.
Shrawan Kumar, Kac–Moody Groups, Their Flag Varieties, and Representation Theory, Birkhäuser, Boston, 2002.
R. P. Langlands and D. Shelstad, On the definition of transfer factors, Mathematische Annalen 278 (1987), 219–271.
Robert Moody and Arturo Pianzola, Lie Algebras with Triangular Decompositions, J. Wiley, New York, 1995.
Robert Moody and K. L. Teo, Tits’ systems with crystallographic Weyl groups, Journal of Algebra 21 (1972), 178–190.
Jun Morita, Commutator relations in Kac–Moody groups, Proceedings of the Japanese Academy 63 (1987), 21–22.
L. J. Rylands, Fast calculation of structure constants, preprint, 2000.
Ichiro Satake, On a theorem of Cartan, Journal of the Mathematical Society of Japan 2 (1951), 284–305.
Jacques Tits, Sur les constants de structure et le théorème d’existence des algèbres de Lie semi-simple, Publications de l’I. H. E. S. 31 (1966), 21–58.
———, Normalisateurs de tores I. Groupes de Coxeter étendus, Journal of Algebra 4 (1966), 96–116.
———, Le problème de mots dans les groupes de Coxeter, Symposia Math. 1 (1968), 175–185.
———, Uniqueness and presentation of Kac–Moody groups over fields, Journal of Algebra 105 (1987), 542–573.
———, Groupes associés aux algèbres de Kac–Moody, Séminaire Bourbaki, exposé 700 (1988).
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Casselman, B. (2014). Structure constants of Kac–Moody Lie algebras. In: Howe, R., Hunziker, M., Willenbring, J. (eds) Symmetry: Representation Theory and Its Applications. Progress in Mathematics, vol 257. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-1590-3_4
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DOI: https://doi.org/10.1007/978-1-4939-1590-3_4
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