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On the Cartan subalgebras of Lie algebras over small fields. (English) Zbl 1037.17019
A self-normalizing nilpotent subalgebra of a Lie algebra is called Cartan subalgebra. Cartan subalgebras play an important role in the structure theory of Lie algebras. In 1967 D. W. Barnes [Math. Z. 101, 350-355 (1967; Zbl 0166.04103)] showed that every finite-dimensional Lie algebra \(L\) has a Cartan subalgebra if the cardinality of the ground field \(F\) of \(L\) is at least \(\dim_F L-1\). In general, the existence of Cartan subalgebras seems to be still an open problem. Nevertheless, in the paper cited above Barnes also showed that every finite-dimensional solvable Lie algebra has a Cartan subalgebra over any field. Since centralizers of maximal tori are Cartan subalgebras, the latter do also exist for finite-dimensional restricted Lie algebras over any field of prime characteristic.
In the paper under review the author shows that any minimal example of a finite-dimensional Lie algebra without Cartan subalgebras must be semisimple. The rest of the paper is devoted to the Cartan subalgebras of the general linear Lie algebras \(\mathfrak{gl}(n,F)\). If the cardinality of \(F\) is larger than \(n^2\), then it is proved that a subalgebra \(H\) of \(\mathfrak{gl}(n,F)\) is a Cartan subalgebra if and only if \(H\) is the centralizer of an element which has exactly \(n\) characteristic roots in some extension field of \(F\). The author also shows for an arbitrary field \(F\) that a subalgebra of \(\mathfrak{gl}(n,F)\) is a Cartan subalgebra if and only if it is a maximal torus. Finally, the author determines all Cartan subalgebras of \(\mathfrak{gl}(2,F)\) for any field \(F\). As a consequence, he obtains that \(\mathfrak{gl}(2,F)\) has \(p^{2n}\) Cartan subalgebras if the cardinality of \(F\) is \(p^n\).

MSC:
17B50 Modular Lie (super)algebras
17B45 Lie algebras of linear algebraic groups
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