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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.

17B50 Modular Lie (super)algebras
17B56 Cohomology of Lie (super)algebras
Full Text: DOI
[1] DOI: 10.1016/0021-8693(69)90051-9 · Zbl 0257.16024
[2] DOI: 10.1090/S0002-9939-1965-0180633-X
[3] DOI: 10.1016/0021-8693(83)90236-3 · Zbl 0521.17003
[4] Farnsteiner R., Proc. Amer. Math. Soc 91 pp 41– (1984)
[5] Herstein I.N., Math. Assn. of America 91 (1968)
[6] Hochschild G.P., Proc. Amer. Math. Soc 5 pp 603– (1954)
[7] DOI: 10.2307/2372701 · Zbl 0055.26505
[8] Jacobson N., Lie Algebras (1979)
[9] DOI: 10.1090/S0002-9904-1965-11300-3 · Zbl 0134.19103
[10] DOI: 10.1016/0021-8693(83)90238-7 · Zbl 0515.17006
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