Relating Kac-Moody, Virasoro and Krichever-Novikov algebras (Q1115955)

Affine Lie algebras may be regarded as spaces of Lie-algebra-valued meromorphic functions on \(P^ 1({\mathbb{C}})\) with poles only at 0 and \(\infty\). Similarly, the Virasoro algebra may be regarded as the meromorphic vector fields on \(P^ 1({\mathbb{C}})\) with poles only at 0 and \(\infty\). \textit{I. M. Krichever} and \textit{S. P. Novikov} [Funkts. Anal. Prilozh. 21, No.4, 47-61 (1987; Zbl 0659.17012)] generalized these algebras to any compact Riemann surface with two distinguished points \(P_{\pm}.\) The present authors show that general Krichever-Novikov algebras are isomorphic to those in the genus 0 case in two ways, corresponding to choosing local parameters at \(P_{\pm}\). In particular, the generalized grading of the Krichever-Novikov algebras is seen to be an artifact of the choice of bases.