4-POINT AFFINE LIE-ALGEBRAS

We consider Lie algebras of the form g X R where g is a simple complex Lie algebra and R = C[s, s(-1), (s(-1))(-1), (s-a)(-1)] for a is an element of C-(0, 1). After showing that R is isomorphic to a quadratic extension of the ring C[t, t(-1)] of Laurent polynomials, we prove that g X R is a quasi-graded Lie algebra with a triangular decomposition. We determine the universal central extension of g X R and show that the cocycles defining it are closely related to ultraspherical (Gegenbauer) polynomials.