In mathematics, in the representation theory of algebraic groups, an observable subgroup is an algebraic subgroup of a linear algebraic group whose every finite-dimensional rational representation arises as the restriction to the subgroup of a finite-dimensional rational representation of the whole group.
An equivalent formulation, in case the base field is closed, is that K is an observable subgroup of G if and only if the quotient variety G/K is a quasi-affine variety.
Some basic facts about observable subgroups:
- Every normal algebraic subgroup of an algebraic group is observable.
- Every observable subgroup of an observable subgroup is observable.
Famous quotes containing the word observable:
“To develop an empiricist account of science is to depict it as involving a search for truth only about the empirical world, about what is actual and observable.... It must involve throughout a resolute rejection of the demand for an explanation of the regularities in the observable course of nature, by means of truths concerning a reality beyond what is actual and observable, as a demand which plays no role in the scientific enterprise.”
—Bas Van Fraassen (b. 1941)