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Characterization of Simple Highest Weight Modules

Published online by Cambridge University Press:  20 November 2018

Volodymyr Mazorchuk and
Kaiming Zhao
Volodymyr Mazorchuk
Affiliation:
Department of Mathematics, Uppsala University, Box 480, SE-751 06, Uppsala, Sweden e-mail: mazor@math.uu.se
Kaiming Zhao
Affiliation:
Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5 e-mail: kzhao@wlu.ca
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Abstract.

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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 algebras and for Heisenberg algebras.

Keywords

17B2017B6517B6617B68Lie algebrahighest weight moduletriangular decompositionlocally nilpotent action

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 3 , 01 September 2013, pp. 606 - 614
Copyright
Copyright © Canadian Mathematical Society 2013

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