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Characterization of Simple Highest Weight Modules
  • Volodymyr Mazorchuk,
    Department of Mathematics, Uppsala University, Box 480, SE-751 06, Uppsala, Sweden
  • Kaiming Zhao,
    Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5, Canada
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Abstract

We prove that for simple complex finite dimensional Lie algebras, affine Kac-Moody Lie algebras, the Virasoro algebra and the Heisenberg-Virasoro algebra, simple highest weight modules are characterized by the property that all positive root elements act on these modules locally nilpotently. We also show that this is not the case for higher rank Virasoro and for Heisenberg algebras.
Keywords: Lie algebra, highest weight module, triangular decomposition, locally nilpotent action Lie algebra, highest weight module, triangular decomposition, locally nilpotent action
MSC Classifications: 17B20, 17B65, 17B66, 17B68 show english descriptions Simple, semisimple, reductive (super)algebras
Infinite-dimensional Lie (super)algebras [See also 22E65]
Lie algebras of vector fields and related (super) algebras
Virasoro and related algebras 17B20 - Simple, semisimple, reductive (super)algebras
17B65 - Infinite-dimensional Lie (super)algebras [See also 22E65]
17B66 - Lie algebras of vector fields and related (super) algebras
17B68 - Virasoro and related algebras
 
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