Linear algebraic groups are, roughly speaking, groups of matrices defined by polynomial equations. In practice, however, they are studied in a rich, intrinsic way that avoids explicit matrix manipulations. These linear algebraic groups arise in an astonishing range of contexts. Linear algebraic groups over finite fields help describe almost all of the finite simple groups, over the real and complex numbers they give a perspective on Lie groups, and over more general fields, such as number fields, they give a very rich source of field theoretic and arithmetic information.

This course will focus on the structure theory of linear algebraic groups over fields in general, with an emphasis on understanding the “classical” groups. We’ll assume some familiarity with algebraic geometry (e.g. at the level of Hartshorne Chapter I), but we will also spend the first couple of weeks with a relatively self-contained review of the necessary facts to make sure that we will share a common language and framework.

The main topics covered will include fundamentals of the theory of smooth linear algebraic groups over general fields (commutators, centralizers, normalizers, Lie algebras, quotients, Jordan decomposition, solvable groups, diagonalizable groups, unipotent groups, the Lie-Kolchin theorem, and the solvable and unipotent radicals). From here, we will attempt to give an overview of root systems and the structure of reductive groups.

Course structure:

There will be light but consistent required work: roughly one problem per week (chosen from a suggested list), on a flexible timeline.

Texts:

While we will not follow any one text in a strict way, this course is heavily inspired by Brian Conrad’s course (https://virtualmath1.stanford.edu/~conrad/252Page/) for which excellent lecture notes are available. Besides this, useful references covering similar material are: Humphreys’ Linear Algebraic Groups; Springer’s Linear Algebraic Groups, and Borel’s Linear Algebraic Groups, and the notes from Brian Conrad’s follow-up course (https://virtualmath1.stanford.edu/~conrad/249BW16Page/).