Central extensions

1. Introduction

Affine Lie algebras are infinite dimensional generalizations of Lie algebras and are special cases of Kac-Moody algebras. They turn out to be extremely useful in describing the symmetry properties of conformal field theories–quantum field theories with conformal invariance. They can be defined in a couple of ways starting from a Lie algebra. One way is to consider the loop algebras associated with a Lie algebra and then consider central extensions of this loop algebra. The second and equivalent way is to define them using the Cartan matrix of a Lie algebra and extending it (by an additional row and column) and then generalize the Serre relations. Recall that a Lie algebra can be defined using its Cartan matrix and Serre relations. Extending Lie algebras in this way would lead to a modified Dynkin diagram where the diagram corresponding to the Lie algebra would have an added root. I plan to discuss both these approaches but in this post, I’ll discuss the central extension point of view as it leads naturally to geometric aspects of affine Lie algebras. This point of view is explained in the book by Pressley and Segal and most of what I have in this post is from this book. I also found notes by Ko Honda useful, which are based on the book by Kohno.