Math 822 Seminar in Algebra - Introduction to Lie Theory

This course is intended as an introduction to the theory of Lie groups and Lie algebras. The founder of Lie theory was the mathematician Sophus Lie (1842-1899), who, more than one hundred years ago, said about his work: "I am certain, absolutely certain that... these theories will be recognized as fundamental at some point in the future". We know now that he was absolutely right, since the notions of Lie groups and Lie algebras are in the vocabulary of every mathematician and physicist today. Lie's theories are indispensable tools for understanding the physical laws of Nature. Important advances have been made since Lie's work, through a better understanding of the subtle interplay between algebra, geometry, and combinatorics related to Lie theory; some of these connections will be discussed in this course. Furthermore, a considerable amount of recent work is devoted to the theory of quantum groups, which are certain deformations connected to Lie groups. Quantum groups are related to other recent developments in mathematics and theoretical physics. However, they will not be discussed in this course.

Tentative syllabus. Definition of a Lie group and of the associated Lie algebra. The root system associated to a semisimple Lie algebra and the corresponding group of reflections (Weyl group). The construction of a semisimple Lie algebra from a root system. The classification of simple Lie algebras. Combinatorics of reflection groups and root systems. Affine Weyl groups. The geometric counterpart: flag varieties, Schubert varieties, their cohomology and K-theory.

About the terminology. Lie groups are sets endowed simultaneously with the compatible structures of a group and of a C^infinity manifold. A typical example is the group of n by n invertible matrices over the complex numbers. To each Lie group one associates its tangent space at the identity, which is naturally endowed with a skew-symmetric bilinear map known as the Lie bracket; this leads to the corresponding Lie algebra. The passage to Lie algebras is motivated by the simpler structure of the latter. Simple Lie algebras are completely classified into several infinite families and a finite number of exceptional ones. The crucial object for this classification is the root system associated to a semisimple Lie algebra, which is just a finite collection of vectors in R^n with strong symmetry properties. The group of reflections in the hyperplanes through the origin perpendicular to the mentioned vectors (roots) is finite; this is the Weyl group. This group is naturally endowed with the structure of a partially ordered set, called the Bruhat order, which has a very interesting combinatorics. There are certain varieties associated with Lie groups, that arise as certain quotients of them; the simplest example is the Grassmannian, that is, the variety of all k-dimensional subspaces of C^n.

Prerequisites: Linear algebra (including eigenvalues, bilinear forms, Euclidean spaces, and tensor products of vector spaces), and some acquaintance with the methods of abstract algebra. Some acquaintance with topology would be helpful during the last part of this class, but is not required. No knowledge of representation theory is required; on the other hand, this class will be useful for people who intend to take my representation theory course in the near future.

Textbooks: Excerpts from the following books will be used (in order of importance):

Cristian Lenart, Department of Mathematics, ES 118, SUNY at Albany