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:
“Every living language, like the perspiring bodies of living creatures, is in perpetual motion and alteration; some words go off, and become obsolete; others are taken in, and by degrees grow into common use; or the same word is inverted to a new sense or notion, which in tract of time makes an observable change in the air and features of a language, as age makes in the lines and mien of a face.”
—Richard Bentley (16621742)