A linear algebraic group is a matrix group that is also an affine variety. In particular, its elements satisfy polynomial equations. The group operations are required to be given by regular rational functions. The linear algebraic groups are similar to the Lie groups, except that linear algebraic groups may be defined over any field, including those of positive field characteristic. The special linear group of matrices of determinant one SL(n) is a linear algebraic group. This is because the equation for the determinant is a polynomial equation in the entries of the matrices. The general linear group of matrices with nonzero determinant GL(n) is also a linear algebraic group.

affine variety | algebraic group | group | Lie algebra | Lie group | variety