Abstract
Infinite many invariant sectors of the regular representation of loop groups are constructed in this Letter. One of these sectors contains the cyclic component of the vacuum Ω0 (the constant vector) of the regular representation. This shows that the regular representation of a loop group does not admit the vacuum Ω0 as a cyclic vector.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Malliavin, M. and Malliavin, P.: Integration on loop groups I. Quasi in variant measures, J. Funct. Anal. 93 (1990), 207–237.
Albeverio, S., Høegh-Krohn, R., and Testard, D.: Irreducibility and reducibility for the energy representation of the group of mappings of a Riemannian manifold into a compact semisimple Lie group, J. Funct. Anal. 41 (1981), 378–396.
Albeverio, S., Høegh-Krohn, R., Marion, J., Testard, D., and Torrésani, B.: Noncommutative Distributions, Marcel Dekker, New York, 1993.
Frenkel, I.: Orbital theory for affine Lie algebras, Invent. Math. 77 (1984), 301–352.
Warner, G.: Harmonic Analysis on Semisimple Lie Groups I, Springer-Verlag, Berlin, Heidelberg, New York, 1972.
Albeverio, S., Høegh-Krohn, R., Testard, D., and Vershik, A.: Factorial representations of path groups, J. Funct. Anal. 51 (1981), 115–131.
Author information
Authors and Affiliations
Additional information
This work was supported in part by the National Natural Science Foundation of China.
Rights and permissions
About this article
Cite this article
Zheng-Dong, W. Invariant sectors of the regular representation of loop groups. Lett Math Phys 37, 461–465 (1996). https://doi.org/10.1007/BF00312676
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00312676