unproven parts, precise references are mentioned.]{}'
address: |
    Departamento de matemática y aplicaciones, Cure\
    Universidad de la República\
    Tacuarembó entre Av. Artigas y Aparicio Saravia\
    CP 20000, Maldonado\
    Uruguay\
    wrferrer@cure.edu.uy
author:
- Walter Ferrer Santos
title: A short survey on observability
---

[^1]

Introduction {#section:intro}
============

The concept of observable subgroup of an affine algebraic group $G$ was introduced by A. Bialynicki–Birula, G. Hochschild and G.D. Mostow in 1963 in [@kn:hmbb]: *Extension of representations of algebraic linear groups* (hereafter referred to as ERA).

Initially the notion of observability was related to the following situation.

Assume that $H \subseteq G$ is a pair of a subgroup and a group. We say that a representation $(V,\rho)$ of $G$ is an extension of a representation $(U,\sigma)$ of $H$ if: $U \subseteq V$ and the action $\rho: G \times V \to V$ restricts to $\sigma:H \times U \to
U$[^2].

The main question adressed by the authors of [@kn:hmbb] concerns the following problem: in the case that $H \text{ and } G$ are affine algebraic groups, and the representations are finite dimensional and rational, does every representation of $H$ admits an extension? In the situation that the answer is positive the group is said to 