Abstract
The Mickelsson-Faddeev extension is a 3-space analogue of a Kac-Moody group, where the central charge is replaced by a space of functions of the gauge potential. This extension is a pullback of a universal extension, where the gauge potentials are replaced by operators in a Schatten ideal, as in non-commutative differential geometry. Our main result is that the universal extension cannot be faithfully represented by unitary operators on a separable Hilbert space. We also examine potential consequences of the existence of unitary representations for the Mickelsson-Faddeev extension.
Similar content being viewed by others
References
de la Harpe, P.: Classical Banach-Lie Algebras and Banach-Lie Groups of Operators in Hilbert Space. Lecture Notes in Mathematics, vol.285, Berlin, Heidelberg, New York: Springer 1972
Mickelsson, J.: Kac-Moody groups and the Dirac determinant line bundle. In: Topological and geometrical methods in field theory, p. 117–131. Hietarinta, J., Westerholms, J. (eds.), Singapore: World Scientific 1986
Mickelsson, J.: Current algebra representation for the 3+1 dimensional Dirac-Yang-Mills theory. Commun. Math. Phys.117, 261–277 (1988)
Mickelsson, J., Rajeev, S. G.: Current algebras ind+1 dimensions and determinant bundles over infinite dimensional Grassmannians. Commun. Math. Phys.116, 365–400 (1988)
Pickrell, D.: Separable representations for automorphism groups of infinite rank symmetric spaces, to appear in J.F.A.
Pickrell, D.: Mackey analysis of infinite semidirect products, submitted to Pac. J. Math.
Pressley, A., and Segal, G.: Loop groups. Oxford: Oxford University Press 1987
Segal, G.: Faddeev's anomaly in Gauss's law, preprint
Connes, A.: Non-commutative differential geometry. Publ. Math. I.H.E.S.62, 257–360 (1986)
Frenkel, I.: Beyond Affine Lie Algebras. Proceedings of the I.C.M., 1986, p. 821
Gelfand, I., Vilenkin, N.: Generalized functions, vol. 4. New York: Academic Press 1964
Author information
Authors and Affiliations
Additional information
Communicated by A. Jaffe
Rights and permissions
About this article
Cite this article
Pickrell, D. On the Mickelsson-Faddeev extension and unitary representations. Commun.Math. Phys. 123, 617–625 (1989). https://doi.org/10.1007/BF01218587
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01218587