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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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