Krichever-Novikov Type Algebras. A General Review and the Genus Zero Case

In the first part of this survey we recall the definition and some of the constructions related to Krichever-Novikov type algebras. Krichever and Novikov introduced them for higher genus Riemann surfaces with two marked points in generalization of the classical algebras of Conformal Field Theory. Schlichenmaier extended the theory to the multi-point situation and even to a larger class of algebras. The almost-gradedness of the algebras and the classification of almost-graded central extensions play an important role in the theory and in applications. In the second part we specialize the construction to the genus zero multi-point case. This yields beside instructive examples also additional results. In particular, we construct universal central extensions for the involved algebras, which are vector field algebras, differential operator algebras, current algebras and Lie superalgebras. We point out that the recently (re-)discussed N-Virasoro algebras are nothing else as multi-point genus zero Krichever-Novikov type algebras. The survey closes with structure equations and central extensions for the three-point case.