Affine Kac-Moody Lie algebras as torsors over the punctured line

Abstract. We interpret and develop a theory of loop algebras as torsors (principal homoge-neous spaces) over Spec (k[t, t−1]). As an application, we recover Kac’s realization of affine Kac-Moody Lie algebras. Introduction. There is a beautiful construction of Victor Kac’s, realizing affine Kac-Moody Lie algebras over the complex numbers as (twisted) loop algebras. The construction gives explicit generators for the algebras, which are then shown to satisfy the relations corresponding to the affine Cartan matrix at hand. In this short note, we propose to look at loop algebras in a completely different way