= Linearly reductive algebraic group
An <affine algebraic group> is linearly reductive when every finite-dimensional <rational representation> is completely reducible. Equivalently its invariants functor is exact. Over a field of <characteristic zero>, <reductive algebraic groups> are linearly reductive; in positive characteristic this stronger property fails for many reductive groups. An <algebraic torus> is linearly reductive in every characteristic because after extension to a separable closure its <rational representations> decompose into character spaces, and exactness of the invariants functor descends to the original <field>.
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