Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 2 (2006), 003, 4 pages      math.PR/0601330      https://doi.org/10.3842/SIGMA.2006.003

Heat Kernel Measure on Central Extension of Current Groups in any Dimension

Rémi Léandre
Institut de Mathématiques de Bourgogne, Université de Bourgogne, 21000 Dijon, France

Received October 30, 2005, in final form January 13, 2006; Published online January 13, 2006

Abstract
We define measures on central extension of current groups in any dimension by using infinite dimensional Brownian motion.

Key words: Brownian motion; central extension; current groups.

pdf (133 kb)   ps (110 kb)   tex (7 kb)

References

  1. Airault H., Malliavin P., Integration on loop groups. II. Heat equation for the Wiener measure, J. Funct. Anal., 1992, V.104, 71-109.
  2. Albeverio S., Hoegh-Krohn R., The energy representation of Sobolev Lie groups, Compos. Math., 1978, V.36, 37-52.
  3. Araki H., Bogoliubov automorphisms and Fock representations of canonical anticommutation relations, in Operator Algebras and Mathematical Physics, Editors P.A.T. Jorgensen and P.S. Muhly, Contemp. Math., 1987, V.62, 23-143.
  4. Baxendale P., Markov processes on manifolds of maps, Bull. Amer. Math. Soc., 1976, V.82, 505-507.
  5. Baxendale P., Wiener processes on manifolds of maps, Proc. Roy. Soc. Edinburgh Sect. A, 1980/1981, V.87, 127-152.
  6. Belopolskaya Y.L., Daletskii Yu.L., Stochastic differential equation and differential geometry, Kluwer, 1990.
  7. Berman S., Billig Y., Irreducible representations for toroidal Lie algebras, J. Algebra, 1999, V.221, 188-231.
  8. Berman S., Gao Y., Krylyuk Y., Quantum tori and the structure of elliptic quasi-simple algebras, J. Funct. Anal., 1996, V.135, 339-389.
  9. Brzezniak Z., Elworthy K.D., Stochastic differential equations on Banach manifolds, Methods Funct. Anal. Topology, 2000, V.6, 43-80.
  10. Brzezniak Z., Léandre R., Horizontal lift of an infinite dimensional diffusion, Potential Anal., 2000, V.12, 43-84.
  11. Carey A.L., Ruijsenaars S.N.M., On fermion gauge groups, current algebras and Kac-Moody algebras, Acta Appl. Math., 1987, V.10, 1-86.
  12. Daletskii Yu.L., Measures and stochastic differential equations on infinite-dimensional manifolds, in Espace de lacets, Editors R. Léandre, S. Paycha and T. Wuerzbacher, Publication University of Strasbourg, 1996, 45-52.
  13. Etingof P., Frenkel I., Central extensions of current groups in two dimension, Comm. Math. Phys., 1994, V.165, 429-444.
  14. Frenkel I., Khesin B., Four dimensional realization of two-dimensional current groups, Comm. Math. Phys., 1996, V.178, 541-562.
  15. Kuo H.H., Diffusion and Brownian motion on infinite dimensional manifolds, Trans. Amer. Math. Soc., 1972, V.159, 439-451.
  16. Léandre R., A unitary representation of the basical central extension of a loop group, Nagoya Math. J., 2000, V.159, 113-124.
  17. Léandre R., Brownian surfaces with boundary and Deligne cohomology, Rep. Math. Phys., 2003, V.52, 353-362.
  18. Léandre R., Galton-Watson tree and branching loops, in Geometry, Integrability and Quantization. VI (Softek), Editors A. Hirshfeld and I. Mladenov, 2005, 276-283.
  19. Léandre R., Measures on central extension of current groups in two dimension, in Infinite Dimensional Analysis (Jagna), Editors C. Bernido and V. Bernido, to appear.
  20. Maier P., Neeb K.H., Central extensions of current group, Math. Ann., 2003, V.326, 367-415.
  21. Mickelsson J., Current algebras and groups, Plenum Press, 1989.
  22. Milnor J., Remarks on infinite-dimensional Lie groups, in Relativity, Groups and Topology II, Editor B. Dewitt, North-Holland, 1984, 1007-1057.
  23. Paris G., Wu Y.S., Perturbation theory without gauge fixing, Sci. Sinica, 1981, V.24, 483-496.
  24. Pressley A., Segal G., Loop groups, Oxford University Press, 1986.
  25. Zhiyang S., Toroidal groups, Comm. Algebra, 1992, V.20, 3411-3458.


Previous article   Next article   Contents of Volume 2 (2006)