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Deformations of the Witt, Virasoro, and Current Algebra

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Generalized Lie Theory in Mathematics, Physics and Beyond

For finite-dimensional Lie algebras there is a tight connection between cohomology theory and deformations of the Lie algebra. This is not the case anymore if the Lie algebra to be deformed is infinite dimensional. Such Lie algebras might be formally rigid but nevertheless allow deformations which are even locally non-trivial. In joint work with Alice Fialowski the author constructed such geometric families for the formally rigid Witt algebra and current Lie algebras. These families are genus one (i.e. elliptic) Lie algebras of Krichever—type. In this contribution the results are reviewed. The families of algebras are given in explicit form. The constructions are induced by the geometric process of degenerating the elliptic curves to singular cubics.

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References

  1. Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B 241, 333–380 (1984)

    Article  MathSciNet  Google Scholar 

  2. Fialowski, A.: An example of formal deformations of Lie algebras. NATO Conference on Deformation Theory of Algebras and Applications, Proceedings, Kluwer 1988, 375–401

    Google Scholar 

  3. Fialowski, A.: Deformations of some Infinite Dimensional Lie Algebras. J. Math. Phys. 31, 1340–1343 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  4. Fialowski, A., Fuchs, D.: Construction of Miniversal Deformations of Lie Algebras. J. Funct. Anal. 161, 76–110 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Fialowski, A., Schlichenmaier, M.: Global Deformations of the Witt algebra of Krichever—Novikov Type. Comm. Contemp. Math. 5, 921–945 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Fialowski, A., Schlichenmaier, M.: Global Geometric Deformations of Current Algebras as Krichever—Novikov Type Algebras. Comm. Math. Phys. 260, 579–612 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Gerstenhaber, M.: On the Deformation of Rings and Algebras I,II,III Ann. Math. 79, 59–10 (1964); 84, 1–19 (1966); 88, 1–34 (1968)

    MathSciNet  Google Scholar 

  8. Krichever I.M., Novikov S.P.:Algebras of Virasoro type, Riemann surfaces and structures of the theory of solitons. Funktional Anal. i. Prilozhen. 21, 46–63 (1987); Virasoro type algebras, Riemann surfaces and strings in Minkowski space. Funktional Anal. i. Prilozhen. 21, 47–61 (1987); Algebras of Virasoro type, energy-momentum tensors and decompositions of operators on Riemann surfaces, Funktional Anal. i. Prilozhen. 23, 46–63 (1989)

    MathSciNet  Google Scholar 

  9. Lecomte, P., Roger, C.: Rigidity of Current Lie Algebras of Complex Simple Type. J. London Math. Soc. (2) 37, 232–240 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  10. Nijenhuis, A., Richardson, R.: Cohomology and Deformations of Algebraic Structures. Bull.Amer. Math. Soc. 70, 406–411 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  11. Nijenhuis, A., Richardson, R.: Cohomology and Deformations in Graded Lie Algebras. Bull.Amer. Math. Soc. 72, 1–29 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  12. Schlichenmaier, M.: Verallgemeinerte Krichever - Novikov Algebren und deren Darstellungen,University of Mannheim, June 1990

    Google Scholar 

  13. Schlichenmaier, M.: Krichever—Novikov algebras for more than two points. Lett. Math. Phys.19, 151–165 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  14. Schlichenmaier, M.: Krichever—Novikov algebras for more than two points: explicit generators. Lett. Math. Phys. 19, 327–336 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  15. Schlichenmaier, M.: Central Extensions and Semi-Infinite Wedge Representations of Krichever-Novikov Algebras for More than Two Points. Lett. Math. Phys. 20, 33–46 (1991)

    Article  MathSciNet  Google Scholar 

  16. Schlichenmaier, M.: Degenerations of Generalized Krichever—Novikov Algebras on Tori. Jour.Math. Phys. 34, 3809–3824 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  17. Schlichenmaier, M.: Differential operator algebras on compact Riemann surfaces. Generalized Symmetries in Physics (Clausthal 1993, Germany) (H.-D. Doebner, V.K. Dobrev, A.G. Ushveridze, eds.), World Scientific, Singapore, 1994, pp. 425–434

    Google Scholar 

  18. Schlichenmaier, M.: Local Cocycles and Central Extensions for Multi-Point Algebras of Krichever—Novikov Type. J. reine angew. Math. 559, 53–94 (2003)

    MATH  MathSciNet  Google Scholar 

  19. Schlichenmaier, M.: Higher Genus Affine Lie algebras of Krichever—Novikov Type, Moscow Math. Jour. 3, 1395–142 (2003)

    MATH  MathSciNet  Google Scholar 

  20. Schlichenmaier, M., Sheinman, O.K.: The Sugawara construction and Casimir operators for Krichever-Novikov algebras. J. Math. Sci., New York 92, no. 2, 3807–3834 (1998), q-alg/9512016

    Article  MATH  MathSciNet  Google Scholar 

  21. Schlichenmaier, M., Sheinman, O.: Wess-Zumino-Witten-Novikov Theory, Knizhnik-Zamolodchikov Equations, and Krichever—Novikov Algebras. Russian Math. Surv. 54, 213–250 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  22. Schlichenmaier, M., Sheinman, O.: Knizhnik–Zamolodchikov Equations for Positive Genus and Krichever—Novikov Algebras. Russian Math. Surv. 59, 737–770 (2004)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, University of Luxembourg, Campus Limpertsberg, 162 A, Avenue de la Faiencerie, Luxembourg, Grand-Duchy of Luxembourg, 1511

    M. Schlichenmaier

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  1. M. Schlichenmaier
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Correspondence to M. Schlichenmaier .

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Editors and Affiliations

  1. Centre for Mathematical Sciences, Division of Mathematics, Lund Institute of Technology, Lund University, Box 118, Lund, 221 00, Sweden

    Sergei Silvestrov

  2. Department of Mathematics, Tallinn University of Technology, Ehitajate tee 5, Tallinn, 19086, Estonia

    Eugen Paal

  3. Institute of Pure Mathematics, University of Tartu, J. Liivi 2, Tartu, 50409, Estonia

    Viktor Abramov

  4. Department of Mathematical Sciences, Mathematical Sciences Chalmers University of Technology and Göteborg University, Göteborg, 412 96, Sweden

    Alexander Stolin

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Schlichenmaier, M. (2009). Deformations of the Witt, Virasoro, and Current Algebra. In: Silvestrov, S., Paal, E., Abramov, V., Stolin, A. (eds) Generalized Lie Theory in Mathematics, Physics and Beyond. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85332-9_19

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