Hilbert's Theorem 90 for non-compact groups
Bull. Korean Math. Soc. 2017 Vol. 54, No. 5, 1757-1771
Published online July 17, 2017
Printed September 30, 2017
Marat Rovinsky
National Research University Higher School of Economics
Abstract : Let $K$ be a field and $G$ be a group of its automorphisms. It follows from Speiser's generalization of Hilbert's Theorem 90, \cite{Speiser} that any $K$-{\it semilinear} representation of the group $G$ is isomorphic to a direct sum of copies of $K$, if $G$ is finite. In this note three examples of pairs $(K,G)$ are presented such that certain irreducible $K$-semilinear representations of $G$ admit a simple description: (i) with precompact $G$, (ii) $K$ is a field of rational functions and $G$ permutes the variables, (iii) $K$ is a universal domain over field of characteristic zero and $G$ its automorphism group. The example (iii) is new and it generalizes the principal result of \cite{adm}.
Keywords : non-compact groups
MSC numbers : 14C15, 14F20, 14F43, 20B27, 20C32
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