Lectures on Kac-Moody Algebras with Applications in (Super-)Gravity

These lectures are divided into two main parts. In the first part we give an introduction to the theory of Kac-Moody algebras directed towards physicists. In particular, we describe the subclasses of affine and Lorentzian Kac-Moody algebras in detail. Our treatment focuses on the Chevalley-Serre presentation, and emphasizes the importance of the Weyl group. We illustrate the basic theory with simple examples. In the second part of the lectures we make use of the contents of part one to describe some recent developments devoted to investigations of the underlying symmetry structures of supergravity theories. We begin by describing how toroidal compactifications of gravity theories reveals hidden global and local symmetries of the reduced Lagrangian. We also discuss attempts at extending these symmetry structures to infinite-dimensional algebras. We show how a manifestly Kac-Moody invariant action can be constructed, whose solutions correspond to exact BPS solutions in maximal supergravity theories.