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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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