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CHARACTER CLUSTERS FOR LIE ALGEBRA MODULES OVER A FIELD OF NONZERO CHARACTERISTIC
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 89 / Issue 2 / April 2014
- Published online by Cambridge University Press:
- 07 June 2013, pp. 234-242
- Print publication:
- April 2014
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- Article
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- You have access
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For a Lie algebra
$L$ over an algebraically closed field 
$F$ of nonzero characteristic, every finite dimensional 
$L$-module can be decomposed into a direct sum of submodules such that all composition factors of a summand have the same character. Using the concept of a character cluster, this result is generalised to fields which are not algebraically closed. Also, it is shown that if the soluble Lie algebra 
$L$ is in the saturated formation 
$\mathfrak{F}$ and if 
$V, W$ are irreducible 
$L$-modules with the same cluster and the 
$p$-operation vanishes on the centre of the 
$p$-envelope used, then 
$V, W$ are either both 
$\mathfrak{F}$-central or both 
$\mathfrak{F}$-eccentric. Clusters are used to generalise the construction of induced modules.
