Linear algebraic group

In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I where M^T is the transpose of M.

The main examples of linear algebraic groups are certain Lie groups, where the underlying field is the real or complex field. (For example, every compact Lie group can be regarded as the group of points of a real linear algebraic group, essentially by the Peter–Weyl theorem.) These were the first algebraic groups to be extensively studied. Such groups were known for a long time before their abstract algebraic theory was developed according to the needs of major applications. Compact Lie groups were considered by Élie Cartan, Ludwig Maurer, Wilhelm Killing, and Sophus Lie in the 1880s and 1890s in the context of differential equations and Galois theory. However, a purely algebraic theory was first developed by Kolchin (1948), with Armand Borel as one of its pioneers. Picard–Vessiot theory party inspired algebraic groups (giving the archaic term "Vessiot variety" for "linear algebraic group").

One of the first uses for the theory was to define the Chevalley groups.

...
Wikipedia