MA453 Content

Content:

Lie algebras are a natural class of algebraic objects which occur in many areas of mathematics and physics. The Lie algebras considered in this course will be finite dimensional vector spaces endowed with a new multiplication, called a Lie bracket, which is almost never associative. We start by introducing these objects and studying some of their basic properties. A key example is the vector space of $n$ by $n$ matrices with the Lie bracket $[A,B] = AB - BA$ . We move on to study nilpotent and soluble Lie algebras in detail, proving Engel's and Lie's theorems. The remainder of the course builds towards a central goal: classifying the simple complex Lie algebras. To do this we need to study the adjoint representation, the Lie algebra $\mathfrak{sl}_2$ and its representation theory.

Books:

J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, 1979

N. Jacobson, Lie Algebras, Dover, 1979

K.Erdmann and M. Wildon, Introduction to Lie Algebras, Springer 2006