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

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Harmonic and Complex Analysis and its Applications

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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 is given in geometric terms. Some applications, including the semi-infinite wedge form representations are recalled. We close by giving some remarks on the Lax operator algebras introduced recently by Krichever and Sheinman.

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Notes

  1. 1.

    In the book [18] arguments are given why it is more appropriate just to use Virasoro algebra, as Witt introduced “his” algebra in a characteristic p context. Nevertheless, I decided to stick here to the most common convention.

  2. 2.

    Here δ k l is the Kronecker delta which is equal to 1 if k = l, otherwise zero.

  3. 3.

    Strictly speaking, there are some special cases where some constants have to be added such that the Krichever–Novikov duality (39) is valid, see [36].

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Acknowledgements

Partial support by the Internal Research Project GEOMQ11, University of Luxembourg, is acknowledged.

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  1. Mathematics Research Unit, University of Luxembourg, FSTC, Campus Kirchberg, 6, rue Coudenhove-Kalergi, 1359, Luxembourg-Kirchberg, Luxembourg

    Martin Schlichenmaier

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    Alexander Vasil'ev

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Schlichenmaier, M. (2014). From the Virasoro Algebra to Krichever–Novikov Type Algebras and Beyond. In: Vasil'ev, A. (eds) Harmonic and Complex Analysis and its Applications. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-01806-5_7

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