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Unitary representations of the maps S1 → su(N, 1)

Published online by Cambridge University Press:  24 October 2008

Vyjayanthi Chari  and
Andrew Pressley
Vyjayanthi Chari
Affiliation:
School of Mathematics, The Institute for Advanced Study, Princeton, NJ 08540, U.S.A.
Andrew Pressley
Affiliation:
Department of Mathematics, King's College, London WC2R 2LS
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Extract

For many questions, both in Mathematics and in Physics, the most important representations of a Lie algebra a are those which are unitarizable and highest weight (such representations are automatically irreducible). The classification of such representations when a is a finite-dimensional complex simple Lie algebra was completed only recently (see [3] for details and further references) and the corresponding question when a is an affine algebra was investigated by Jakobsen and Kac [5]. Theorem 3·1 of that paper contains a list of unitarizable highest weight representations which is claimed to be exhaustive. However, we shall show that this list is incomplete by constructing a further family of such representations. In fact, the classification problem in the affine case must be considered to be still open.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 102 , Issue 2 , September 1987 , pp. 259 - 272
Copyright
Copyright © Cambridge Philosophical Society 1987

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References

REFERENCES

[1]Chari, V.. Integrable representations of affine Lie algebras. Invent. Math. 85 (1986), 317–335.CrossRefGoogle Scholar
[2]Chari, V. and Pressley, A. N.. New unitary representations of loop groups. Math. Ann. 275 (1986), 87–104.Google Scholar
[3]Enright, T., Howe, R. and Wallach, N.. A classification of unitary highest weight modules. In Representation theory of reductive groups (ed. Trombi, P. C.), Progress in Math. vol. 40 (Birkhäuser, 1983).Google Scholar
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[5]Jakobsen, H. and Kac, V.. A new class of unitarizable highest weight representations of infinite-dimensional Lie algebras. In Non-linear equations in quantum field theory, Lecture Notes in Physics, vol. 226 (Springer-Verlag, 1985), 1–20.Google Scholar
[6]Kac, V.. Infinite dimensional Lie algebras. Progress in Math. vol. 44 (Birkhäuser, 1983).CrossRefGoogle Scholar