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Central extensions of groups of sections

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Abstract

If K is a Lie group and q : P → M is a principal K-bundle over the compact manifold M, then any invariant symmetric V-valued bilinear form on the Lie algebra \({\mathfrak{k}}\) of K defines a Lie algebra extension of the gauge algebra by a space of bundle-valued 1-forms modulo exact 1-forms. In this article, we analyze the integrability of this extension to a Lie group extension for non-connected, possibly infinite-dimensional Lie groups K. If K has finitely many connected components, we give a complete characterization of the integrable extensions. Our results on gauge groups are obtained by the specialization of more general results on extensions of Lie groups of smooth sections of Lie group bundles. In this more general context, we provide sufficient conditions for integrability in terms of data related only to the group K.

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References

  1. Bishop, R.L., Crittenden, R.J.: Geometry of Manifolds. AMS Chelsea Publications, AMS, Rrovidence, Rhode Island (1964)

  2. Bott R.: The space of loops on a Lie group. Mich. Math. J. 5, 35–61 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bourbaki N.: Lie Groups and Lie Algebra, chap. I–III. Springer, Berlin (1989)

    Google Scholar 

  4. Cartan H., Eilenberg S.: Homological Algebra. Princeton University Press, Princeton, NJ (1956)

    MATH  Google Scholar 

  5. Chevalley C., Eilenberg S.: Cohomology theory of Lie groups and Lie algebras. Trans. Am. Math. Soc. 63, 85–124 (1948)

    Article  MATH  MathSciNet  Google Scholar 

  6. Daboul, C., Daboul, J., Slodowy, P.: The dynamical algebra of the hydrogen atom as a twisted loop algebra. In: Group Theoretical Methods in Physics (Toyonaka, 1994), pp. 175–178. World Sci. Publ., River Edge, NJ (1995)

  7. de la Harpe, P.: Classical Banach–Lie Algebras and Banach–Lie Groups of Operators in Hilbert Space. Lecture Notes in Mathematics, vol. 285. Springer-Verlag, Berlin (1972)

  8. de la Harpe, P.: Topics in Geometric Group Theory. Chicago Lectures in Mathematics. The University of Chicago Press, Chicago (2000)

  9. Glöckner, H.: Patched locally convex spaces, almost local mappings, and diffeomorphism groups of non-compact manifolds (2006)

  10. Glöckner, H., Neeb, K.-H.: Infinite-dimensional Lie Groups, vol. I, Basic Theory and Main Examples (book in preparation)

  11. Helgason S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, New York, London (1978)

    MATH  Google Scholar 

  12. Hu S.: Homotopy Theory. Academic Press, New York, London (1959)

    MATH  Google Scholar 

  13. Jacobson N.: Lie Algebras. Dover Publications, New York (1979)

    Google Scholar 

  14. Kac V.G.: Infinite-dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  15. Kassel C., Loday J.-L.: Extensions centrales d’algèbres de Lie. Ann. Inst. Fourier 32(4), 119–142 (1982)

    MATH  MathSciNet  Google Scholar 

  16. Kriegl, A., Michor, P.: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, vol. 53. American Mathematical Society, Providence (1997)

  17. Kuiper N.H.: The homotopy type of the unitary group of Hilbert space. Topology 3, 19–30 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  18. Losev A., Moore G., Nekrasov N., Shatashvili S.: Central extensions of gauge groups revisited. Sel. Math. 4, 117–123 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  19. Mackenzie, K.C.H.: General Theory of Lie Groupoids and Lie Algebroids. London Mathematical Society Lecture Note Series, vol. 213. Cambridge University Press, Cambridge (2005)

  20. Maier P., Neeb K.-H.: Central extensions of current groups. Math. Ann. 326(2), 367–415 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  21. Maissen B.: Über Topologien im Endomorphismenraum eines topologischen Vektorraums. Math. Ann. 151, 283–285 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  22. Mickelsson J.: Current Algebras and Groups. Plenum Press, New York (1989)

    MATH  Google Scholar 

  23. Milnor J.: Remarks on infinite-dimensional Lie groups. In: DeWitt, B., Stora, R. (eds) Relativité, groupes et topologie II (Les Houches, 1983), pp. 1007–1057. North Holland, Amsterdam (1984)

    Google Scholar 

  24. Mimura M.: Homotopy theory of Lie groups. In: James, I.M. (eds) Handbook of Algebraic Topology, North Holland, Amsterdam (1995)

    Google Scholar 

  25. Murray M.K.: Another construction of the central extension of the loop group. Commun. Math. Phys. 116, 73–80 (1988)

    Article  Google Scholar 

  26. Neeb K.-H.: Borel–Weil theory for loop groups. In: Huckleberry, A., Wurzbacher, T. (eds) Infinite Dimensional Kähler Manifolds, DMV-Seminar, vol. 31, Birkhäuser Verlag, Berlin (2001)

    Google Scholar 

  27. Neeb K.-H.: Central extensions of infinite-dimensional Lie groups. Ann. Inst. Fourier 52, 1365–1442 (2002)

    MATH  MathSciNet  Google Scholar 

  28. Neeb K.-H.: Universal central extensions of Lie groups. Acta Appl. Math. 73(1,2), 175–219 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  29. Neeb K.-H.: Classical Hilbert–Lie groups, their extensions and their homotopy groups. In: Strasburger, A., Wojtynski, W., Hilgert, J., Neeb, K.-H. (eds) Geometry and Analysis on Finite and Infinite-dimensional Lie Groups, vol. 55, pp. 87–151. Banach Center Publications, Warszawa (2002)

    Google Scholar 

  30. Neeb K.-H.: Towards a Lie theory of locally convex groups. Jpn. J. Math. (3rd Series) 1(2), 291–468 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  31. Neeb, K.-H.: Lie groups of bundle automorphisms and their extensions. In: Neeb, K.-H., Pianzola, A. (eds.) Trends and Developments in Infinite Dimensional Lie Theory, Progress in Mathematics. Birkhäuser Verlag, Berlin (2009, to appear)

  32. Neeb K.-H., Wagemann F.: The second cohomology of current algebras of general Lie algebras. Can. J. Math. 60(4), 892–922 (2008)

    MATH  MathSciNet  Google Scholar 

  33. Neeb K.-H., Wagemann F.: Lie group structures on groups of smooth and holomorphic maps. Geom. Dedic. 134, 17–60 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  34. Neher, E.: Lectures on Root Graded and Extended Affine Lie Algebras, Version of May 11, 2007

  35. Onishchick A.L., Vinberg E.B.: Lie Groups and Algebraic Groups. Springer-Verlag, Berlin (1990)

    Google Scholar 

  36. Pansu, P.: Superrigidité géométrique et applications harmoniques, math.DG/0612736

  37. Pianzola A., Prelat D., Sun J.: Descent constructions for central extensions of infinite dimensional Lie algebras. Manuscr. Math. 122(2), 137–148 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  38. Pressley A., Segal G.: Loop Groups. Oxford University Press, Oxford (1986)

    MATH  Google Scholar 

  39. Schottenloher, M.: A Mathematical Introduction to Conformal Field Theory. Lecture Notes in Physics, vol. m43. Springer, Berlin (1997)

  40. Vizman C.: The path group construction of Lie group extensions. J. Geom. Phys. 58(7), 860–873 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  41. Wockel C.: Lie group structures on symmetry groups of principal bundles. J. Funct. Anal. 251, 254–288 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  42. Wockel C.: The Samelson Product and Rational Homotopy for Gauge Groups. Abh. Math. Sem. Univ. Hamburg 77, 219–228 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  43. Wockel, C.: A generalisation of Steenrod’s approximation theorem. Arch. Math. (Brno) (to appear)

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Authors and Affiliations

  1. Fachbereich Mathematik, Technische Universität Darmstadt, Schloßgartenstr. 7, 64285, Darmstadt, Germany

    Karl-Hermann Neeb

  2. Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstr. 3–5, 37073, Göttingen, Germany

    Christoph Wockel

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  1. Karl-Hermann Neeb
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  2. Christoph Wockel
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Neeb, KH., Wockel, C. Central extensions of groups of sections. Ann Glob Anal Geom 36, 381–418 (2009). https://doi.org/10.1007/s10455-009-9168-6

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  • DOI: https://doi.org/10.1007/s10455-009-9168-6

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