Reductive group

Summary

A reductive group is a connected linear algebraic group with no nontrivial connected normal unipotent subgroup. Examples are $\mathrm{GL}_n$, $\mathrm{SL}_n$, the special orthogonal groups $\mathrm{SO}_n$, the symplectic groups, and the algebraic tori. Over an algebraically closed field, reductive groups are classified by root data. In characteristic $0$, they are the connected linear groups whose representations are all semisimple.