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 and . For the case of we show that there are no simple torsion-free -modules of degree *k* for and . We conclude with some examples showing that there exist simple torsion-free -modules of degrees , and *n*.

**[BBL]**G. Benkart, D. J. Britten, and F. W. Lemire,*On the tensor product of pointed torsion free and simple finite dimensional**modules*(in progress).**[BL1]**D. J. Britten and F. W. Lemire,*A classification of simple Lie modules having a 1-dimensional weight space*, Trans. Amer. Math. Soc.**299**(1987), 687-697. MR**869228 (88b:17013)****[BL2]**-,*On basic cycles of*and , Canad. J. Math.**XXXVII**(1985), 122-140.**[BL3]**-,*Irreducible representations of**with a 1-dimensional weight space*, Trans. Amer. Math. Soc.**272**(1982), 509-540.**[F]**Suren Fernado,*Lie algebra modules with finite dimensional weight spaces*. I, Trans. Amer. Math. Soc.**322**(1987), 757-781. MR**1013330 (91c:17006)****[J]**J. Humphreys,*Introduction to Lie algebras and representation theory*, Graduate Texts in Math., vol. 9, Springer-Verlag, New York, 1972. MR**0323842 (48:2197)**

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DOI:
https://doi.org/10.1090/S0002-9939-1995-1246518-1

Article copyright:
© Copyright 1995
American Mathematical Society