Universal central extensions of elliptic affine Lie algebras

Abstract

Let 𝔤 be a simple complex (finite dimensional) Lie algebra, and let R be the ring of regular functions on a compact complex algebraic curve with a finite number of points removed. Lie algebras of the form 𝔤⊗CR are considered; these generalize Kac–Moody loop algebras since for a curve of genus zero with two punctures R≂C[t,t−1]. The universal central extension of 𝔤⊗R is analogous to an untwisted affine Kac–Moody algebra. By Kassel’s theorem the kernel of the universal central extension is linearly isomorphic to the Kähler differentials of R modulo exact differentials. The dimension of the kernel for any R is determined first. Restricting to hyperelliptic curves with 2, 3, or 4 special points removed, a basis for the kernel is determined. Restricting further to an elliptic curve with punctures at two points (of orders one and two in the group law) we explicitly determine the cocycles which give the commutation relations for the universal central extension. The results involve Pollaczek polynomials, which are a genus-one generalization of ultraspherical (Gegenbauer) polynomials. © 1994 American Institute of Physics.