nLab
linear algebraic group

A linear algebraic group GG is any algebraic subgroup of GL(n,k)GL(n,k) where kk is a field and nn a natural number.

An algebraic group is linear iff it is affine.

An algebraic group scheme is affine if the underlying scheme is affine.

The category of affine group schemes is the opposite of the category of commutative Hopf algebras.