Four-point affine Lie algebras

Abstract:

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