Hostname: page-component-77c89778f8-n9wrp Total loading time: 0 Render date: 2024-07-19T15:25:21.136Z Has data issue: false hasContentIssue false
  • English
  • Français

Basic representations of some affine Lie algebras and generalized Euler identities

Part of: Combinatorics Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  09 April 2009

Kailash C. Misra
Kailash C. Misra
Affiliation:
Department of MathematicsNorth Carolina State UniversityRaleigh, North Carolina 27695-8205, U. S. A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider certain affine Kac-Moody Lie algebras. We give a Lie theoretic interpretation of the generalized Euler identities by showing that they are associated with certain filtrations of the basic representations of these algebras. In the case when the algebras have prime rank, we also give algebraic proofs of the corresponding identities.

MSC classification

Secondary: 17B65: Infinite-dimensional Lie (super)algebras 05A19: Combinatorial identities, bijective combinatorics
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 42 , Issue 3 , June 1987 , pp. 296 - 311
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Andrews, G. E., The Theory of Partitions (Rota, G. C. (ed.), Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley, Reading, Mass., 1976).Google Scholar
[2]Kac, V. G., Infinite Dimensional Lie Algebras (Progress in Mathematics, Vol. 44, Birkhäuser, Boston, Mass., 1983).CrossRefGoogle Scholar
[3]Kac, V. G., Kazhdam, D. A., Lepowsky, J. and Wilson, R. L., ‘Realization of the basic represntations of the Euclidean Lie algebras’, Advances in Math. 2 (1981), 83–112.CrossRefGoogle Scholar
[4]Lepowsky, J. and Wilson, R. L., ‘A Lie Theoretic interpretation and proof of the Rogers-Ramanujan identities’, Advances in Math. 45 (1982), 21–72.CrossRefGoogle Scholar
[5]Lepowsky, J. and Wilson, R. L., ‘The structure of standard modules, I: Universal algebras and the Rogers-Ramanujan identities’, Invent. Math. 77 (1984), 199–290.Google Scholar
[6]Lepowsky, J. and Wilson, R. L., ‘The structure of standard modules, II: The case A(1)1, principal gradation’, Invent. Math. 79 (1985), 417–442.CrossRefGoogle Scholar
[7]Misra, K. C., ‘Structure of certain standard modules for As(1)n and the Rogers-Ramanujan identities’, J. Algebra 88 (1984), 196–227.CrossRefGoogle Scholar
[8]Misra, K. C., ‘Structure of some standard modules for C(1)n’, J. Algebra 90 (1984), 385–409.CrossRefGoogle Scholar
[9]Misra, K. C., ‘Standard representations of some affine Lie algebras’ (Lepowsky, J., Mandelstam, S. and Singer, I. M. (eds.), Vertex Operators in Mathematics and Physics, Publ. Math. Sci. Res. Inst., Vol. 3, Springer-Verlag, Berlin and New York, 1985), pp. 163–183.CrossRefGoogle Scholar