Injective modules over restricted Lie algebras (Q1108361)

Let \(L\) be a restricted Lie algebra over a field \(k\) of characteristic \(p\). An \(L\)-module always means a restricted module. The author first proves the following facts. (1) If the trivial one-dimensional module \(k\) is injective, then the \(p\)-map \(x\to x^{[p]}\) is injective. (2) If \(E\) is an injective \(L\)-module, \(S\) a finite-dimensional \(L\)-module, then \(S\oplus_ k E\) is injective. As a consequence, Hochschild's necessary and sufficient conditions for every \(L\)-module to be completely reducible [cf. \textit{G. Hochschild}, Proc. Am. Math. Soc. 5, 603--605 (1954; Zbl 0056.03102)] are found. The author also obtains the following results: If \(L\) is finite-dimensional and (1) if trivial modules are injective, then \(L\) is abelian and the \(p\)-map is surjective; (2) if \(L\) is abelian and the \(p\)-map is surjective, then every \(L\)-module is injective. (3) If \(k\) is algebraically closed (\(L\) may be infinite-dimensional), then every simple \(L\)-module is injective if and only if \(L\) is abelian with basis \(\{e_{\alpha}\}\) such that \(e_{\alpha}^{[p]}=e_{\alpha}\).