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Mosc. Math. J., 2003, Volume 3, Number 4, Pages 1395–1427 (Mi mmj136)  

This article is cited in 28 scientific papers (total in 28 papers)

Higher genus affine algebras of Krichever–Novikov type

M. Schlichenmaier

University of Luxembourg

Abstract: For higher genus multi-point current algebras of Krichever–Novikov type associated to a finite-dimensional Lie algebra, local Lie algebra two-cocycles are studied. They yield as central extensions almost-graded higher genus affine Lie algebras. In case that the Lie algebra is reductive a complete classification is given. For a simple Lie algebra, like in the classical situation, there is up to equivalence and rescaling only one non-trivial almost-graded central extension. The classification is extended to the algebras of meromorphic differential operators of order less or equal one on the currents algebras.

Key words and phrases: Krichever–Novikov algebras, central extensions, almost-grading, conformal field theory, infinite-dimensional Lie algebras, affine algebras, differential operator algebras, local cocycles.


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MSC: 17B67, 17B56, 17B66, 14H55, 17B65, 30F30, 81R10, 81T40
Received: October 24, 2002

Citation: M. Schlichenmaier, “Higher genus affine algebras of Krichever–Novikov type”, Mosc. Math. J., 3:4 (2003), 1395–1427

Citation in format AMSBIB
\by M.~Schlichenmaier
\paper Higher genus affine algebras of Krichever--Novikov type
\jour Mosc. Math.~J.
\yr 2003
\vol 3
\issue 4
\pages 1395--1427

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    This publication is cited in the following articles:
    1. Schlichenmaier M.T., “Local cocycles and central extensions for multipoint algebras of Krichever-Novikov type”, J Reine Angew Math, 559 (2003), 53–94  crossref  mathscinet  zmath  isi
    2. O. K. Sheinman, “Projective Flat Connections on Moduli Spaces of Riemann Surfaces and the Knizhnik–Zamolodchikov Equations”, Proc. Steklov Inst. Math., 251 (2005), 293–304  mathnet  mathscinet  zmath
    3. Fialowski A., Schlichenmaier M., “Global geometric deformations of current algebras as Krichever-Novikov type algebras”, Comm. Math. Phys., 260:3 (2005), 579–612  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. Sheinman O.K., “Krichever-Novikov algebras and their representations”, Noncommutative Geometry and Representation Theory in Mathematical Physics, Contemporary Mathematics Series, 391, 2005, 313–321  crossref  mathscinet  zmath  isi
    5. I. M. Krichever, O. K. Sheinman, “Lax Operator Algebras”, Funct. Anal. Appl., 41:4 (2007), 284–294  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    6. O. K. Sheinman, “Krichever–Novikov Algebras, their Representations and Applications in Geometry and Mathematical Physics”, Proc. Steklov Inst. Math., 274, suppl. 1 (2011), S85–S161  mathnet  crossref  crossref  zmath
    7. Fialowski A., Schlichenmaier M., “Global geometric deformations of the Virasoro algebra, current and affine algebras by Krichever-Novikov type algebras”, International Journal of Theoretical Physics, 46:11 (2007), 2708–2724  crossref  mathscinet  zmath  adsnasa  isi
    8. Hartwig B., Terwilliger P., “The Tetrahedron algebra, the Onsager algebra, and the sl(2) loop algebra”, Journal of Algebra, 308:2 (2007), 840–863  crossref  mathscinet  zmath  isi
    9. Schlichenmaier M., “Higher Genus Affine Lie Algebras of Krichever - Novikov Type”, Difference Equations, Special Functions and Orthogonal Polynomials, 2007, 589–599  crossref  mathscinet  zmath  isi
    10. Schlichenmaier M., “A global operator approach to Wess-Zumino-Novikov-Witten models”, XXVI Workshop on Geometrical Methods in Physics, AIP Conference Proceedings, 956, 2007, 107–119  crossref  mathscinet  zmath  adsnasa  isi
    11. M. Schlichenmaier, O. K. Sheinman, “Central extensions of Lax operator algebras”, Russian Math. Surveys, 63:4 (2008), 727–766  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    12. O. K. Sheinman, “Lax Operator Algebras and Integrable Hierarchies”, Proc. Steklov Inst. Math., 263 (2008), 204–213  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    13. Ito T., Terwilliger P., “Finite-Dimensional Irreducible Modules for the Three-Point 2 Loop Algebra”, Communications in Algebra, 36:12 (2008), 4557–4598  crossref  mathscinet  zmath  isi
    14. Neeb K.-H., Wagemann F., “Lie group structures on groups of smooth and holomorphic maps on non-compact manifolds”, Geometriae Dedicata, 134:1 (2008), 17–60  crossref  mathscinet  zmath  isi
    15. Schlichenmaier M., “Classification of central extensions of Lax operator algebras”, Geometric Methods in Physics, AIP Conference Proceedings, 1079, 2008, 227–234  crossref  mathscinet  zmath  adsnasa  isi
    16. Schlichenmaier M., “Deformations of the Witt, Virasoro, and Current Algebra”, Generalized Lie Theory in Mathematics, Physics and Beyond, 2009, 219–234  crossref  mathscinet  zmath  isi
    17. Cox B., Futorny V., “Djkm Algebras I: their Universal Central Extension”, Proc Amer Math Soc, 139:10 (2011), 3451–3460  crossref  mathscinet  zmath  isi
    18. M. Schlichenmaier, “Multipoint Lax operator algebras: almost-graded structure and central extensions”, Sb. Math., 205:5 (2014), 722–762  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    19. Knibbeler V., Lombardo S., Sanders J.A., “Automorphic Lie Algebras With Dihedral Symmetry”, J. Phys. A-Math. Theor., 47:36 (2014), 365201  crossref  mathscinet  zmath  isi  elib
    20. Cox B., Jurisich E., “Realizations of the Three-Point Lie Algebra Sl(2, R) Circle Plus (Omega(R)/Dr)”, Pac. J. Math., 270:1 (2014), 27–47  crossref  mathscinet  isi  elib
    21. Cox B., Guo X., Lu R., Zhao K., “N-Point Virasoro Algebras and Their Modules of Densities”, Commun. Contemp. Math., 16:3 (2014), 1350047  crossref  mathscinet  zmath  isi
    22. Schlichenmaier M., “Krichever-Novikov Type Algebras: Theory and Applications”, Krichever-Novikov Type Algebras: Theory and Applications, Degruyter Studies in Mathematics, 53, Walter de Gruyter Gmbh, 2014, 1–360  crossref  mathscinet  isi
    23. Cox B., Jurisich E., Martins R.A., “the 3-Point Virasoro Algebra and Its Action on a Fock Space”, J. Math. Phys., 57:3 (2016), 031702  crossref  mathscinet  zmath  isi
    24. Schlichenmaier M., “N-Point Virasoro Algebras Considered as Krichever-Novikov Type Algebras”, Geometric Methods in Physics, Trends in Mathematics, eds. Kielanowski P., Ali S., Bieliavsky P., Odzijewicz A., Schlichenmaier M., Voronov T., Springer Int Publishing Ag, 2016, 295–308  crossref  mathscinet  zmath  isi
    25. Cox B., Guo X., Lu R., Zhao K., “Simple Superelliptic Lie Algebras”, Commun. Contemp. Math., 19:3 (2017), 1650032  crossref  mathscinet  zmath  isi
    26. Schlichenmaier M., “N-Point Virasoro Algebras Are Multipoint Krichever-Novikov-Type Algebras”, Commun. Algebr., 45:2 (2017), 776–821  crossref  mathscinet  zmath  isi
    27. Knibbeler V., Lombardo S., Sanders J.A., “Higher-Dimensional Automorphic Lie Algebras”, Found. Comput. Math., 17:4 (2017), 987–1035  crossref  zmath  isi  scopus
    28. Cox B., Zhao K., “Certain Families of Polynomials Arising in the Study of Hyperelliptic Lie Algebras”, Ramanujan J., 46:2 (2018), 323–344  crossref  mathscinet  zmath  isi  scopus
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