From the Virasoro Algebra to Krichever--Novikov Type Algebras and Beyond

Abstract: Starting from the Virasoro algebra and its relatives the generalization to higher genus compact Riemann surfaces was initiated by Krichever and Novikov. The elements of these algebras are meromorphic objects which are holomorphic outside a finite set of points. A crucial and non-trivial point is to establish an almost-grading replacing the honest grading in the Virasoro case. Such an almost-grading is given by splitting the set of points of possible poles into two non-empty disjoint subsets. Krichever and Novikov considered the two-point case. Schlichenmaier studied the most general multi-point situation with arbitrary splittings. Here we will review the path of developments from the Virasoro algebra to its higher genus and multi-point analogs. The starting point will be a Poisson algebra structure on the space of meromorphic forms of all weights. As sub-structures the vector field algebras, function algebras, Lie superalgebras and the related current algebras show up. All these algebras will be almost-graded. In detail almost-graded central extensions are classified. In particular, for the vector field algebra it is essentially unique. The defining cocycle are given in geometric terms. Some applications, including the semi-infinite wedge form representations are recalled. Finally, some remarks on the by Krichever and Sheinman recently introduced Lax operator algebras are made.