
TL;DR
This survey reviews the historical development and key concepts of observability in algebraic group actions, highlighting its significance in invariant theory, representation theory, and related mathematical areas.
Contribution
It provides a comprehensive overview of the evolution, definitions, and main results of observability, including recent extensions like observable actions and adjunctions.
Findings
Historical development of observability concepts
Introduction of observable actions and adjunctions
Connections to invariant and representation theory
Abstract
The exploration of the notion of observability exhibits transparently the rich interplay between algebraic and geometric ideas in \emph{geometric invariant theory}. The concept of \emph{observable subgroup} was introduced in the early 1960s with the purpose of studying extensions of representations from an affine algebraic subgroup to the whole group. The extent of its importance in \emph{representation and invariant theory} in particular for Hilbert's problem was noticed almost immediately. An important strenghtening appeared in the mid 1970s when the concept of \emph{strong observability} was introduced and it was shown that the notion of observability can be understood as an intermediate step in the notion of reductivity (or semisimplicity), when adequately generalized. More recently starting in 2010, the concept of observable subgroup was expanded to include the…
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