UNIVERSAL CENTRAL EXTENSIONS OF ELLIPTIC AFFINE LIE-ALGEBRAS

Let g be a simple complex (finite dimensional) Lie algebra, acid 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 gx(C )R are considered; these generalize Kac-Moody loop algebras since for a curve of genus zero with two punctures R similar or equal to C[t,t(- 1)]. The universal central extension of gxR is analogous to an untwist ed affine Kac-Moody algebra. By Kassel's-theorem the kernel of the uni versal central extension is linearly isomorphic to the Kahler differen tials of R module 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 point s (of orders one and two in the group law) we explicitly determine the cocycles which give the commutation relations for the universal centr al extension. The results involve Pollaczek polynomials, which are a g enus-one generalization of ultraspherical (Gegenbauer) polynomials. (C ) 1994 American Institute of Physics.