STRUCTURE AND REPRESENTATIONS OF CONFORMAL ALGEBRAS

0.1. In the last two decades vertex algebras have been an important tool in such diverse subjects as representations of infinite-dimensional algebras and the theory of finite groups [15, 19]. Roughly speaking, a vertex algebra is a space V such that to each element of V there corresponds a formal distribution, i.e. an element of EndV [[z±1]]. (Note that any algebraic operation performed on the space of formal distribution will have to involve more than one variable, as the distributions are power series in both z and z−1.) Two distributions a(z) and b(w) must be local, that is, commute outside the diagonal of the zw-plane.