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Cohomology of restricted Lie algebras. (English) Zbl 0055.26505

A restricted Lie algebra \(L\), over a field of characteristic \(p\), is defined to be [cf. N. Jacobson, Trans. Am. Math. Soc. 50, 15–25 (1941; Zbl 0025.30301)] a Lie algebra over \(F\) in which there exists a mapping \(x\to x^{[p]}\) of \(L\) into itself which satisfies certain identities. The author defines restricted cohomology groups of such algebras and interprets the 1- and 2-dimensional groups in terms of extensions of restricted \(L\)-modules (\(L\)-modules \(M\) such that \(x^p\cdot m = x^{[p]}\cdot m\) for all \(x\in L\), \(m\in M\)) and of restricted Lie algebras. Throughout he shows the relations between the “restricted” situations and the situations arising in the interpretation of the ordinary cohomology groups of the algebras.

MSC:

17B56 Cohomology of Lie (super)algebras
17B50 Modular Lie (super)algebras
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