Abstract
Let \(\mathfrak {g}\) be a finite dimensional nilpotent p-restricted Lie algebra over a field k of characteristic p. For \(p\geqslant 5\), we show that every endotrivial \(\mathfrak {g}\)-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 \(p \geqslant 5\) 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.
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Acknowledgements
David J. Benson and Jon F. Carlson would like to thank the University of Bielefeld for its hospitality while this work was carried out. Jon F. Carlson is grateful to the Mathematisches Forschungsinstitut Oberwolfach and to the Simons Foundation for financial support.
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Jon F. Carlson was partially supported by Simons Foundation Grant 054813-01.
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Benson, D.J., Carlson, J.F. Endotrivial modules for nilpotent restricted Lie algebras. Arch. Math. 114, 503–513 (2020). https://doi.org/10.1007/s00013-019-01426-2
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DOI: https://doi.org/10.1007/s00013-019-01426-2