2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62873Multi-point algebras of Krichever Novikov type for higher genus Riemann surfaces are generalisations of the Virasoro algebra and its related algebras. Complete existence and uniqueness results for local 2-cocycles defining almost-graded central extensions of the functions algebra, the vector field algebra, and the differential operator algebra (of degree \le 1) are shown. This is applied to the higher genus, multi-point affine algebras to obtain uniqueness for almost-graded central extensions of the current algebra of a simple finite-dimensional Lie algebra. An earlier conjecture of the author concerning the central extension of the differential operator algebra induced by the semi-infinite wedge representations is proved.38 pages, Amslatex, some minor changes in Section 7Quantum AlgebraMathematical PhysicsAlgebraic GeometryRings and Algebras17B66, 17B56, 17B67, 14H55, 17B65, 30F30, 81R10, 81T40text