Journal of Lie Theory

no. 4
Galois Groups and Group Actions on Lie Algebras
A. L. Agore1, 2; G. Militaru3
1Simion Stoilow Inst. of Mathematics, Romanian Academy of Sciences, P.O. Box 1-764, 014700 Bucharest, Romania
2Faculty of Engineering, Vrije Universiteit, Pleinlaan 2, 1050 Brussels, Belgium
3Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei 14, 010014 Bucharest 1, Romania
Journal of Lie Theory, Volume 28 (2018) no. 4, pp. 1165-1188
If $\mathfrak{g} \subseteq \mathfrak{h}$ is an extension of Lie algebras over a field $k$ such that ${\rm dim}_k (\mathfrak{g}) = n$ and ${\rm dim}_k (\mathfrak{h}) = n + m$, then the Galois group ${\rm Gal} \, (\mathfrak{h}/\mathfrak{g})$ is explicitly described as a subgroup of the canonical semidirect product of groups ${\rm GL} (m, \, k) \rtimes {\rm M}_{n\times m} (k)$. An Artin type theorem for Lie algebras is proved: if a group $G$ whose order is invertible in $k$ acts as automorphisms on a Lie algebra $\mathfrak{h}$, then $\mathfrak{h}$ is isomorphic to a skew crossed product $\mathfrak{h}^G \, \#^{\bullet} \, V$, where $\mathfrak{h}^G$ is the subalgebra of invariants and $V$ is the kernel of the Reynolds operator. The Galois group ${\rm Gal} \, (\mathfrak{h}/\mathfrak{h}^G)$ is also computed, highlighting the difference from the classical Galois theory of fields where the corresponding group is $G$. The counterpart for Lie algebras of Hilbert's Theorem 90 is proved and based on it the structure of Lie algebras $\mathfrak{h}$ having a certain type of action of a finite cyclic group is described. Radical extensions of finite dimensional Lie algebras are introduced and it is shown that their Galois group is solvable. Several applications and examples are provided.
DOI: 10.5802/jolt.1044
Classification: 17B05, 17B40
Keywords: Groups acting on Lie algebras, Galois groups, Artin's theorem, Hilbert's theorem 90

A. L. Agore  1 , 2 ; G. Militaru  3

1 Simion Stoilow Inst. of Mathematics, Romanian Academy of Sciences, P.O. Box 1-764, 014700 Bucharest, Romania
2 Faculty of Engineering, Vrije Universiteit, Pleinlaan 2, 1050 Brussels, Belgium
3 Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei 14, 010014 Bucharest 1, Romania 
A. L. Agore; G. Militaru. Galois Groups and Group Actions on Lie Algebras. Journal of Lie Theory, Volume 28 (2018) no. 4, pp. 1165-1188. doi: 10.5802/jolt.1044
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     title = {Galois {Groups} and {Group} {Actions} on {Lie} {Algebras}},
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