Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: From the Virasoro Algebra to Krichever--Novikov Type Algebras and Beyond   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.

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The categories of this paper are: nucl-ex
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The categories of this paper are: stat
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The categories of this paper are: cmp-lg
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The categories of this paper are: ao-sci
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The categories of this paper are: nucl-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: cs
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-fin
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The categories of this paper are: nlin
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The categories of this paper are: cond-mat
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The categories of this paper are: supr-con
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The categories of this paper are: alg-geom
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The categories of this paper are: q-alg