N-Point Virasoro Algebras Are Multipoint Krichever-Novikov-Type Algebras

We show how the recently again discussed N-point Witt, Virasoro, and ane Lie algebras are genus zero examples of the multipoint versions of Krichever-Novikov-type algebras as introduced and studied by Schlichenmaier. Using this more general point of view, useful structural insights and an easier access to calculations can be obtained. The concept of almost-grading will yield information about triangular decompositions which are of importance in the theory of representations. As examples, the algebra of functions, vector fields, differential operators, current algebras, ane Lie algebras, Lie superalgebras, and their central extensions are studied. Very detailed calculations for the three-point case are given.