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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: [email protected]
Kaiming Zhao
Affiliation:
Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5 e-mail: [email protected]
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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
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 3 , 01 September 2013 , pp. 606 - 614
Copyright
Copyright © Canadian Mathematical Society 2013

References

[1] Arbarello, E., Concini, C. De, Kac, V. G., and Procesi, C., Moduli spaces of curves and representation theory. Comm. Math. Phys. 117 (1988), no. 1, 1–36. http://dx.doi.org/10.1007/BF01228409 Google Scholar
[2] Bekkert, V., Benkart, G., and Futorny, V., Weight modules for Weyl algebras. In: Kac-Moody Lie algebras and related topics, Contemp. Math., 343, American Mathematical Society, Providence, RI, 2004, pp. 17–42,Google Scholar
[3] Bekkert, V., Benkart, G., Futorny, V., and Kashuba, I., New irreducible modules for Heisenberg and affine Lie algebras. arxiv:1107.0893.Google Scholar
[4] Dixmier, J., Enveloping algebras. Graduate Studies in Mathematics, 11, American Mathematical Society, Providence, RI, 1996.Google Scholar
[5] Fernando, S., Lie algebra modules with finite-dimensional weight spaces. I. Trans. Amer. Math. Soc. 322 (1990), no. 2, 757–781. http://dx.doi.org/10.2307/2001724 Google Scholar
[6] Futorny, V., Representations of affine Lie algebras. Queen's Papers in Pure and Applied Mathematics, 106, Queen's University, Kingston, ON, 1997.Google Scholar
[7] Hu, J., Wang, X., and Zhao, K., Verma modules over generalized Virasoro algebras Vir[G]. J. Pure Appl. Algebra 177 (2003), no. 1, 61–69. http://dx.doi.org/10.1016/S0022-4049(0200173-1 Google Scholar
[8] Kac, V. G. and Kazhdan, D., Structure of representations with highest weight of infinite-dimensional Lie algebras. Adv. in Math. 34 (1979), no. 1, 97–108. http://dx.doi.org/10.1016/0001-8708(7990066-5 Google Scholar
[9] Kashiwara, M., The universal Verma module and the b-function. In: Algebraic groups and related topics (Kyoto/Nagoya, 1983), Adv. Stud. Pure Math., 6, North-Holland, Amsterdam, 1985, pp. 67–81.Google Scholar
[10] Mazorchuk, V., Verma modules over generalized Witt algebras. Compositio Math. 115 (1999), no. 1, 21–35. http://dx.doi.org/10.1023/A:1000531924778 Google Scholar
[11] Mazorchuk, V., Lectures on sl2(C)-modules. Imperial College Press, London, 2010.Google Scholar
[12] McConnell, J., On the global and Krull dimensions of Weyl algebras over affine coefficient rings. J. London Math. Soc. (2) 29 (1984), no. 2, 249–253. http://dx.doi.org/10.1112/jlms/s2-29.2.249 Google Scholar
[13] Moody, R. and Pianzola, A., Lie algebras with triangular decompositions. Canadian Mathematical Society Series of Monographs and Advanced Texts, JohnWiley & Sons, Inc., New York, 1995.Google Scholar
[14] Patera, J. and Zassenhaus, H., The higher rank Virasoro algebras. Comm. Math. Phys. 136 (1991), no. 1, 1–14. http://dx.doi.org/10.1007/BF02096787 Google Scholar
[15] Shapovalov, N., On bilinear form on the universal enveloping algebra of a complex semisimple Lie algebra. Funct. Anal. Appl. 6 (1972), 307–312.Google Scholar
[16] Zassenhaus, H., Uber Lie’sche Ringe mit Primzahlcharakteristik. Abh. Math. Sem. Univ. Hamburg 13 (1940), 1–100.Google Scholar
[17] Zhelobenko, D. P., S-algebras and Verma modules over reductive Lie algebras. (Russian) Dokl. Akad. Nauk SSSR 273 (1983), no. 4, 785–788.Google Scholar