Abstract
It is shown that all the irreducible integrable highest-weight modules for a Kac-Moody algebra of type C (1)ℓ , and a large class of those for type D (1)ℓ , can be constructed by essentially the same formalism as that employed for certain fundamental modules by Frenkel, Kac-Peterson and Feingold-Frenkel. The idea is to replace the Clifford algebra (or Weyl algebra) by higher analogues of the even Clifford algebra, and the spin, half-spin, or oscillator module by a suitable simple module for this (locally finite) algebra. A submodule of the resulting completely reducible module for the Kac-Moody algebra then affords the desired representation. Properties of the larger module, and analogy with cases previously investigated, suggested the conjecture that our submodule is (often) not proper. In general, this is not the case, as is seen by comparison of characters for certain “basic” examples. The determination of the character of our module rests on finding explicit bases.
1980 AMS subject classification (1985 revision)
- 17B67
- 17B65
Research at Yale University and I.H.E.S., Bures-sur-Yvette, supported in part by National Science Foundation Grants Nos. MCS82-01333 and DMS-8512904.
This paper is in final form, and no version of it will be submitted elsewhere.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Feingold, A. and Frenkel, I., Classical affine algebras. Advances in Math., 58 (1985), 117–172.
Frenkel, I., Spinor representations of affine Lie algebras. Proc. N.A.S. (USA) 77 (1980), 6303–6306.
____, Two constructions of affine Lie algebra representations and boson-fermion correspondence in quantum field theory. J. Funct. Analysis 44 (1981), 259–327.
Kac, V., Infinite Dimensional Lie Algebras, Birkhauser, Boston-Basel-Stuttgart, 1983. (2nd ed., Cambridge Univ. Press, 1985.)
Kac, V. and Peterson, D., Spin and wedge representations of infinite-dimensional Lie algebras and groups. Proc. N.A.S. (USA) 78 (1981), 3308–3312.
Seligman, G., Spin-like modules for certain infinite-dimensional Lie algebras. Séminaire d'algèbre P. Dubreil et M-P. Malliavin, 1985. Springer Lecture Notes in Math. 1220 (1986), pp. 33–55.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1989 Springer-Verlag
About this paper
Cite this paper
Seligman, G.B. (1989). Kac-Moody modules and generalized Clifford algebras. In: Benkart, G., Osborn, J.M. (eds) Lie Algebras, Madison 1987. Lecture Notes in Mathematics, vol 1373. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0088892
Download citation
DOI: https://doi.org/10.1007/BFb0088892
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-51147-2
Online ISBN: 978-3-540-46170-8
eBook Packages: Springer Book Archive
