Anders Buch
Linear Algebraic Groups
A linear algebraic group is an affine variety with a group structure; every such group is a Zariski closed subgroup of a general linear group GL(n). The theory of linear algebraic groups is analogous to the theory of compact Lie groups, but is based on algebraic geometry, which makes it possible to work over fields of positive characteristic. Any connected reductive linear algebraic group is uniquely determined by its root datum, consisting of two dual root systems embedded in dual lattices. We will discuss the ingenious methods used to carve out this information, starting from any linear algebraic group. For example, the roots are weights of the action of a maximal torus on the tangent space at the identity, and each root determines a root subgroup isomorphic to SL(2) or PSL(2), which in turn provides a coroot in the form of the maximal torus of the root subgroup. Before we get there we must first study many preliminary topics, such as the action of a group on its tangent space and coordinate ring, Jordan decomposition in arbitrary linear algebraic groups, the structure of commutative algebraic groups, quotients by closed subgroups, and the existence and properties of maximal tori and Borel subgroups, including Borel's fixed point theorem.
T. A. Springer, Linear Algebraic Groups (2nd edition)
16:640:551 (16:640:353 is recommended).