Loop Algebras and the Virasoro Algebra

Abstract

In this section we introduce some basic theory of loop algebras and related subjects. The reason for doing so is partly to give a useful frame for the Virasoro algebra, which is considered in the next section, and partly to obtain knowledge on the loop group, which will be studied in a later chapter. We notice that the diffeomorphism group Diff⁺(S¹) acts as a group of automorphisms on loop groups and algebras and that the central extension of the Lie algebra Vect(S¹) of Diff⁺(S¹) indeed is the Virasoro algebra. Moreover, the central extension of the loop algebra, together with the further extension discussed below, provide a nice example of Kac-Moody algebras, in fact, they are so-called affine Kac-Moody algebras (see [K-R,p.93–98], [P-S,p.76–78] and [Mi,p.21–23]). Futhermore, the simplest representation of the Kac-Moody algebras is given in terms of the spin representation, for details we refer to [Ar,p.124] (see also [Ve,p.1]). Hereby the connection between the spin representation and the loop algebra is clarified. There is an analogue connection between the metaplectic representation and the Virasoro algebra; however, the analogue construction is more cumbersome [Ve,p.1]. In Chap. 5 of applications we will show, in detail, how these subjects are related for some particular cases.