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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


An analogue of the Jacobson-Morozov theorem for Lie algebras of reductive groups of good characteristics

Author: Alexander Premet
Journal: Trans. Amer. Math. Soc. 347 (1995), 2961-2988
MSC: Primary 17B10; Secondary 17B50, 20G05
MathSciNet review: 1290730
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Abstract: Let $ \mathfrak{g}$ be the Lie algebra of a connected reductive group $ G$ over an algebraically closed field of characteristic $ p > 0$. Suppose that $ {G^{(1)}}$ is simply connected and $ p$ is good for the root system of $ G$. Given a one-dimensional torus $ \lambda \subset G$ let $ \mathfrak{g}(\lambda ,1)$ denote the weight component of $ {\text{Ad(}}\lambda {\text{)}}$ corresponding to weight $ i \in X(\lambda ) \cong \mathbb{Z}$. It is proved in the paper that, for any nonzero nilpotent element $ e \in \mathfrak{g}$, there is a one-dimentional torus $ {\lambda _e} \subset G$ such that $ e \in \mathfrak{g}({\lambda _e},2)$ and $ {\text{Ker}}{\text{ad}}e \subseteq { \oplus _{i \geqslant 0}}\mathfrak{g}({\lambda _e},i)$.

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PII: S 0002-9947(1995)1290730-7
Keywords: Nilpotent element, Lie algebra, reductive group, prime characteristic
Article copyright: © Copyright 1995 American Mathematical Society