Realizations of the Four Point Affine Lie Algebra

for x, y ∈ g, f, g ∈ R, and ω ∈ R/dR, where ( · , · ) denotes the Killing form on g. Here a denotes the image of a ∈ R in the quotient  1 R/dR. A somewhat vague (due to the underspecified R and the resulting imprecisely described basis of R/dR) but natural question arises as to whether there exists free field or Wakimoto type realizations of these algebras. The answer is well known from the work of M. Wakimoto [1986] when R is the ring of Laurent polynomials in one variable. We answer this question below when g = sl(2,C) and R = C[t, t−1, u | u2= t2−2bt+1] is the four point algebra. H. P. Jakobsen and V. Kac [1985] have related work on sl(2, R), and we review the relevant material in Section 7. Before we begin, we mention a little of the genesis of four point algebras. In Kazhdan and Luszig’s explicit study [1993; 1991] of the tensor structure of modules for affine Lie algebras, the ring of functions regular everywhere but a finite number of points appears naturally. M. Bremner named this algebra the n-point algebra. In particular, consider now the Riemann sphere C ∪ {∞} with coordinate function s, and fix four distinct points a1, a2, a3, a4 on it. Let R denote the ring of rational functions with poles only in the set {a1, a2, a3, a4}. The