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

Rémi Léandre

Symmetry, integrability and geometry: methods and applications, Tome 2 (2006)

Résumé

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

Détail

Keywords: Brownian motion; central extension; current groups.

Bibliographie

[1] Airault H., Malliavin P., “Integration on loop groups. II. Heat equation for the Wiener measure”, J. Funct. Anal., 104 (1992), 71–109 | DOI | MR | Zbl

[2] Albeverio S., Hoegh-Krohn R., “The energy representation of Sobolev Lie groups”, Compos. Math., 36 (1978), 37–52 | MR

[3] Araki H., “Bogoliubov automorphisms and Fock representations of canonical anticommutation relations”, Operator Algebras and Mathematical Physics, Contemp. Math., 62, eds. P. A. T. Jorgensen and P. S. Muhly, 1987, 23–143 | MR

[4] Baxendale P., “Markov processes on manifolds of maps”, Bull. Amer. Math. Soc., 82 (1976), 505–507 | DOI | MR | Zbl

[5] Baxendale P., “Wiener processes on manifolds of maps”, Proc. Roy. Soc. Edinburgh Sect. A, 87 (1980/1981), 127–152 | MR | Zbl

[6] Belopolskaya Y. L., Daletskii Yu. L., Stochastic differential equation and differential geometry, Kluwer, 1990 | Zbl

[7] Berman S., Billig Y., “Irreducible representations for toroidal Lie algebras”, J. Algebra, 221 (1999), 188–231 | DOI | MR | Zbl

[8] Berman S., Gao Y., Krylyuk Y., “Quantum tori and the structure of elliptic quasi-simple algebras”, J. Funct. Anal., 135 (1996), 339–389 | DOI | MR | Zbl

[9] Brzezniak Z., Elworthy K. D., “Stochastic differential equations on Banach manifolds”, Methods Funct. Anal. Topology, 6 (2000), 43–80 | MR

[10] Brzezniak Z., Léandre R., “Horizontal lift of an infinite dimensional diffusion”, Potential Anal., 12 (2000), 43–84 | DOI | MR

[11] Carey A. L., Ruijsenaars S. N. M., “On fermion gauge groups, current algebras and Kac–Moody algebras”, Acta Appl. Math., 10 (1987), 1–86 | DOI | MR | Zbl

[12] Daletskii Yu. L., “Measures and stochastic differential equations on infinite-dimensional manifolds”, Espace de lacets, eds. 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., 165 (1994), 429–444 | DOI | MR | Zbl

[14] Frenkel I., Khesin B., “Four dimensional realization of two-dimensional current groups”, Comm. Math. Phys., 178 (1996), 541–562 | DOI | MR | Zbl

[15] Kuo H. H., “Diffusion and Brownian motion on infinite dimensional manifolds”, Trans. Amer. Math. Soc., 159 (1972), 439–451 | DOI | MR

[16] Léandre R., “A unitary representation of the basical central extension of a loop group”, Nagoya Math. J., 159 (2000), 113–124 | MR | Zbl

[17] Léandre R., “Brownian surfaces with boundary and Deligne cohomology”, Rep. Math. Phys., 52 (2003), 353–362 | DOI | MR | Zbl

[18] Léandre R., “Galton–Watson tree and branching loops”, Geometry, Integrability and Quantization, VI, eds. A. Hirshfeld and I. Mladenov, Softek, 2005, 276–283 | MR | Zbl

[19] Léandre R., “Measures on central extension of current groups in two dimension”, Methods and applications of infinite dimensional analysis, eds. C. Bernido and V. Bernido, Cent. Visayan Inst. Found., Jagna, 2006, 172–175 | MR

[20] Maier P., Neeb K. H., “Central extensions of current group”, Math. Ann., 326 (2003), 367–415 | DOI | MR | Zbl

[21] Mickelsson J., Current algebras and groups, Plenum Press, 1989 | MR | Zbl

[22] Milnor J., “Remarks on infinite-dimensional Lie groups”, Relativity, Groups and Topology, II, ed. B. Dewitt, North-Holland, 1984, 1007–1057 | MR

[23] Paris G., Wu Y. S., “Perturbation theory without gauge fixing”, Sci. Sinica, 24 (1981), 483–496 | MR

[24] Pressley A., Segal G., Loop groups, Oxford University Press, 1986 | MR | Zbl

[25] Zhiyang S., “Toroidal groups”, Comm. Algebra, 20 (1992), 3411–3458 | DOI | MR