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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A constraint on the existence of simple torsion free Lie modules


Authors: Daniel Britten, Frank Lemire and Vahid Tarokh
Journal: Proc. Amer. Math. Soc. 123 (1995), 2315-2321
MSC: Primary 17B10
DOI: https://doi.org/10.1090/S0002-9939-1995-1246518-1
MathSciNet review: 1246518
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Abstract: For any simple Lie algebra L with Cartan subalgebra H the classification of all simple H-diagonalizable L-modules having a finite-dimensional weight space is known to depend on determining the simple torsion-free L-modules of finite degree. It is further known that the only simple Lie algebras which admit simple torsion-free modules of finite degree are those of types $ {A_n}$ and $ {C_n}$. For the case of $ {A_n}$ we show that there are no simple torsion-free $ {A_n}$-modules of degree k for $ n \geq 4$ and $ 2 \leq k \leq n - 2$. We conclude with some examples showing that there exist simple torsion-free $ {A_n}$-modules of degrees $ 1,n - 1$, and n.


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DOI: https://doi.org/10.1090/S0002-9939-1995-1246518-1
Article copyright: © Copyright 1995 American Mathematical Society