Linear algebraic groups, such as the general linear group GL_N, the special orthogonal group SO_N or the symplectic group SP_N, can be introduced in a number of different ways. In this course, we will take a functorial approach to the study of linear algebraic groups (more generally, affine group schemes) equivalent to the study of Hopf algebras. The classical view of an algebraic group as a variety will come up as a special case of a smooth algebraic group scheme. Our algebraic approach will be independent of (even complementary to) the analytic approach taken in traditional courses on Lie groups. The first part of the course will follow an excellent introductory text by W. Waterhouse.

• Group schemes and Hopf algebras, and translating between the two
• Representations: modules vs. comodules
• Examples and special cases: abelian group schemes and Cartier duality, etale group schemes, matrix groups, groups of multiplicative type (tori), unipotent groups, nilpotent and solvable groups (i.e. Borel subgroups of semi-simple groups)
• Lie algebras
• Geometric properties: connectedness, irreducibility, smoothness
• Faithful flatness, quotients, and factor groups

Texts:
Introduction to Affine Group Schemes by W. Waterhous (recommended),
Linear Algebraic Groups by J. Humphreys (optional),
The book of Involutions by M-A Knus, A. Merkurjev, M. Rost, and J-P Tignol (optional).

Prerequisites: Math 504/5/6. The course will be suitable for a 2nd year or above graduate student leaning towards an algebra-related field (understood broadly: combinatorics, representation theory, algebraic geometry, algebraic topology). The book by Waterhouse is pretty much self-contained, with an appendix covering the necessary fundamental facts from algebra.