Central extensions of current groups

Maier P, Neeb KH (2003)

Publication Type: Journal article, Original article

Publication year: 2003

Journal

Mathematische Annalen Springer Verlag (Germany)

Publisher: Springer Verlag (Germany)

Book Volume: 326

Pages Range: 367-415

Journal Issue: 2

DOI: 10.1007/s00208-003-0425-x

Abstract

 In this paper we study central extensions of the identity component G of the Lie group C ∞ (M,K) of smooth maps from a compact manifold M into a Lie group K which might be infinite-dimensional. We restrict our attention to Lie algebra cocycles of the form ω(ξ,η)=[κ(ξ,dη)], where κ:𝔨×𝔨→Y is a symmetric invariant bilinear map on the Lie algebra 𝔨 of K and the values of ω lie in Ω1(M,Y)/dC ∞ (M,Y). For such cocycles we show that a corresponding central Lie group extension exists if and only if this is the case for M=𝕊1. If K is finite-dimensional semisimple, this implies the existence of a universal central Lie group extension of G. The groups Diff(M) and C ∞ (M,K) act naturally on G by automorphisms. We also show that these smooth actions can be lifted to smooth actions on the central extension if it also is a central extension of the universal covering group G˜ of G.

Authors with CRIS profile

Karl Hermann Neeb Lehrstuhl für Mathematik (Lie-Gruppen und Darstellungstheorie)

Involved external institutions

Technische Universität Darmstadt DE Germany (DE)

How to cite

APA:

Maier, P., & Neeb, K.H. (2003). Central extensions of current groups. Mathematische Annalen, 326(2), 367-415. https://doi.org/10.1007/s00208-003-0425-x

MLA:

Maier, Peter, and Karl Hermann Neeb. "Central extensions of current groups." Mathematische Annalen 326.2 (2003): 367-415.

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