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Galois groups and group actions on Lie algebras. (arXiv:1505.07346v3 [math.RA] UPDATED)
来源于:arXiv
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 isinvertible 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 查看全文>>