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Kac-Moody modules and generalized Clifford algebras

Part of the Lecture Notes in Mathematics book series (LNM,volume 1373)

Abstract

It is shown that all the irreducible integrable highest-weight modules for a Kac-Moody algebra of type C (1) , and a large class of those for type D (1) , 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.

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References

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© 1989 Springer-Verlag

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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

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  • DOI: https://doi.org/10.1007/BFb0088892

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51147-2

  • Online ISBN: 978-3-540-46170-8

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