Endotrivial modules for nilpotent restricted Lie algebras

Let g be a finite dimensional nilpotent p -restricted Lie algebra over a field k of characteristic p . For p ⩾ 5 , we show that every endotrivial g -module is a direct sum of a syzygy of the trivial module and a projective module. The proof includes a theorem that the intersection of the maximal lin...

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Veröffentlicht in:Archiv der Mathematik 2020-05, Vol.114 (5), p.503-513
Hauptverfasser: Benson, David J., Carlson, Jon F.
Format: Artikel
Sprache:eng
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Zusammenfassung:Let g be a finite dimensional nilpotent p -restricted Lie algebra over a field k of characteristic p . For p ⩾ 5 , we show that every endotrivial g -module is a direct sum of a syzygy of the trivial module and a projective module. The proof includes a theorem that the intersection of the maximal linear subspaces of the null cone of a nilpotent restricted p -Lie algebra for p ⩾ 5 has dimension at least two. We give an example to show that the statement about endotrivial modules is false in characteristic two. In characteristic three, another example shows that our proof fails, and we do not know a characterization of the endotrivial modules in this case.
ISSN:0003-889X
1420-8938
DOI:10.1007/s00013-019-01426-2