A short survey on observability

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 $14^{\text{th}}$ 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 concept of \emph{observable action} of an affine algebraic group on an affine variety, launching a series of new applications. In 2006 the related concept of \emph{observable adjunction} was introduced, and its application to module categories over tensor categories was noticed. In the current survey, we follow (approximately) the historical development of the subject introducing along the way, the definitions and some of the main results including some of the proofs. For the unproven parts, precise references are mentioned.