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Endotrivial modules for nilpotent restricted Lie algebras. (English) Zbl 1442.17019

Given a finite group of order divisible by a prime \(p\), and \(k\) a field of characteristic \(p\), a finitely generated \(kG\)-module is endotrivial if its \(k\)-endomorphism ring is isomorphic to the trivial module in the stable module category.
The notion of an endotrivial module has been extended to other algebraic structures, and in the present paper the authors study the endotrivial modules for finite dimensional nilpotent restricted Lie algebras. If \(\mathfrak g\) is a finite dimensional nilpotent restricted Lie algebra over a field \(k\) of characteristic \(p\), then a \(\mathfrak g\)-module \(M\) is endotrivial if its \(k\)-endomorphism ring is isomomorphic to the direct sum of the trivial \(\mathfrak g\)-module \(k\) and a projective module. Let \(\Omega(M)\) denote the kernel of a projective cover of a \(\mathfrak g\)-module \(M\), and \(\Omega^{-1}(M)\) the cokernel of an injective hull of \(M\). Define iteratively \(\Omega^n(k)=\Omega(\Omega^{n-1}(k))\) and \(\Omega^{-n}(M)=\Omega^{-1}(\Omega^{1-n}(k))\) for all \(n\geq2\).
The authors prove in particular that if \(\mathfrak g\) is a commutative nilpotent restricted \(p\)-Lie algebra, then every endotrivial \(\mathfrak g\)-module is isomorphic to the direct sum of some \(\Omega^n(k)\) plus a projective module.
Their main result in the paper states that if \(p\geq5\) and \(\mathfrak g\) is a nilpotent restricted Lie algebra over a field \(k\) of characteristic \(p\), and if \(M\) is an endotrivial module for \(\mathfrak g\), then \(M\cong\Omega^n(k)\oplus P\), for some integer \(n\in\mathbb Z\) and for some projective module \(P\). The proof of the main theorem involves an examination of the restricted null cone of \(\mathfrak g\).
The authors also show that the result does not extend to characteristic \(2\) by providing an example. If \(p=3\), the authors give an example for which the proof of their main theorem fails.

MSC:

17B50 Modular Lie (super)algebras
20C20 Modular representations and characters
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