# reductive

Let $G$ be a Lie group or algebraic group. $G$ is called reductive over a field $k$ if every representation of $G$ over $k$ is completely reducible

For example, a finite group is reductive over a field $k$ if and only if its order is not divisible by the characteristic of $k$ (by Maschke’s theorem). A complex Lie group is reductive if and only if it is a direct product of a semisimple group and an algebraic torus.

Title reductive Reductive 2013-03-22 13:23:49 2013-03-22 13:23:49 bwebste (988) bwebste (988) 6 bwebste (988) Definition msc 22C05