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Conditions for the commutativity of restricted Lie algebras. (English) Zbl 0565.17005
Let \((L,[p])\) be a restricted Lie algebra of arbitrary dimension over a field \(F\) of positive characteristic \(p\). In the paper an elementary proof is given of one of Jacobson’s conjectures in a special case. The conjecture is the following: every \(L\) satisfying the requirement \(x^{[p]^{n(x)}}=x\) for every \(x\in L\) is commutative.
The main result of the paper is
Theorem 2.3: Let \(F_ p\) denote the union of all Galois fields contained in \(\bar F\). Suppose that \(G\subset L\) is a quasi-toral Lie algebra (an element \(x\in L\) is quasi-toral if there is a positive integer \(n\) such that \(x^{[p]^ n}=x)\). Then \(G\) is abelian if and only if \(G\otimes_ F F_ p\) is quasi-toral.
In Section 3 some cohomological aspects are investigated and the results of J. P. May [Bull. Am. Math. Soc. 71, 372–377 (1965; Zbl 0134.19103)] are generalized to infinite dimension and are applied to quasi-toral Lie algebras.

MSC:
17B50 Modular Lie (super)algebras
17B56 Cohomology of Lie (super)algebras
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References:
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[2] DOI: 10.1090/S0002-9939-1965-0180633-X
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