Abstract
Motivated by positive energy representations , we classify those continuous central extensions of the compactly supported gauge Lie algebra that are covariant under a 1-parameter group of transformations of the base manifold.
B. Janssens acknowledges support from the NWO grant 613.001.214 “Generalised Lie algebra sheaves”.
K.-H. Neeb acknowledges support from the Centre Interfacultaire Bernoulli (CIB) and the NSF (National Science Foundation) for a research visit at the EPFL.
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Notes
- 1.
Note that non-isomorphic maximal ideals of \(\mathfrak {K}_{x}\) are always in different connected components of \(\widehat{M}\), whereas isomorphic maximal ideals may or may not be in the same connected component, depending on the bundle structure.
References
Albeverio, S., Høegh-Krohn, R.J., Marion, J.A., Testard, D.H., Torresani, B.S.: "Noncommutative Distributions - Unitary representations of Gauge Groups and Algebras," Pure and Applied Mathematics 175. Marcel Dekker, New York (1993)
Goldin, G. A.: Lectures on diffeomorphism groups in quantum physics. In: Contemporary Problems in Mathematical Physics, Proceedings of the third international workshop (Cotonue, 2003), pp. 3–93 2004
Janssens, B., Wockel, C.: Universal central extensions of gauge algebras and groups. J. Reine Angew. Math. 682, 129–139 (2013)
Janssens, B., Neeb, K.-H.: Projective unitary representations of infinite dimensional Lie groups. Kyoto J. Math. (2017a). arXiv:1501.00939
Janssens, B., Neeb, K.-H.: Positive energy representations of gauge groups, in preparation (2017b)
Kac, V.: Infinite Dimensional Lie Algebras. Cambridge University Press, Cambridge (1985)
Neeb, K.-H.: Towards a Lie theory of locally convex groups. Jpn. J. Math. 3rd ser. 1:2, 291–468 (2006)
Neeb, K.-H., Wockel, C.: Central extensions of groups of sections. Ann. Glob. Anal. Geom. 36(4), 381–418 (2009)
Pressley, A., Segal, G.: Loop Groups. The Clarendon Press, Oxford University Press, New York (1986)
Tits, J.: Liesche Gruppen und Algebren. Springer, Berlin (1983)
Walter, B., Weighted diffeomorphism groups of Banach spaces and weighted mapping groups, Diss. Math. (Rozprawy Mat.) 484, 128 (2012)
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Janssens, B., Neeb, KH. (2017). Covariant Central Extensions of Gauge Lie Algebras. In: Albeverio, S., Cruzeiro, A., Holm, D. (eds) Stochastic Geometric Mechanics . CIB-SGM 2015. Springer Proceedings in Mathematics & Statistics, vol 202. Springer, Cham. https://doi.org/10.1007/978-3-319-63453-1_6
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DOI: https://doi.org/10.1007/978-3-319-63453-1_6
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