  These are the notes for a Part III course given in the University of
Cambridge in autumn 1998. They contain an exposition of the representation
theory of the Lie algebras of compact matrix groups, affine Kac-Moody algebras
and the Virasoro algebra from a unitary point of view. The treatment uses many
of the methods of conformal field theory, in particular the Goddard-Kent-Olive
construction and the Kazami-Suzuki supercharge operator, a generalisation of
the Dirac operator. The proof of the Weyl character formula is taken from
unpublished notes of Peter Goddard. The supersymmetric proof of the Kac
character formula for affine Kac-Moody algebras was also found independently at
roughly the same time by Greg Landweber in his Harvard Ph.D. dissertation. One
of the main novelties of this approach is a very rapid proof of the
Feigin-Fuchs character formula for the discrete series representations of the
Virasoro algebra. It relies only on the first part of the Friedan-Qiu-Shenker
unitarity theorem placing restrictions on the parameter h. Their result, which
only uses elementary properties of plane curves, is presented in detail in
these notes along with the somewhat harder result for c. A substantially
shorter treatment is possible if the final goal is just the Feigin-Fuchs
formula, since in that case only the representation theory of affine sl(2) is
needed (cf the MSRI summer course given in 2000,
http://www.msri.org/publications/ln/msri/2000/opal/wassermann/). The same
method has been applied by S. Palcoux to the Neveu-Schwarz algebra and also
works for the N=2 superconformal algebra.
