Central extensions of infinite-dimensional Lie groups
The central result of this extensive paper is the following exact sequence for a connected Lie group \(G\), its universal covering group \(\widetilde G\), the discrete central subgroup \(\pi_1(G) \subseteq \widetilde G\), and an abelian Lie group \(Z\) which can be written as \(Z=z/ \Gamma\), where \(\Gamma \subseteq z\) is a discrete subgroup: \[ \begin{multlined} \Hom(G,Z) \hookrightarrow \Hom(\widetilde G,Z)\to \Hom\bigl(\pi_1(G), Z\bigr)\to\text{Ext}_{\text{Lie}} (G,Z)\to\\ \to H^2_c(g,z) \to\Hom\bigl( \pi_2(G),Z\bigr) \times \Hom\bigl(\pi_1(G),\text{Lin} \bigl(g,z)\bigr),\end{multlined} \tag{1} \] where \(\text{Lin} (g,z)\) denotes the space of continuous linear maps \(g\to z\). The author describes the course of the paper as follows: ``In Section 2 we collect the necessary results on central extensions of topological groups and in Section 3 we provide some results on infinite-dimensional manifolds and Lie groups which are well-known in the finite-dimensional case. In Section 4 we explain how the setting for abstract, resp., topological groups has to be modified to deal with central extensions of Lie groups with smooth local sections. Section 5 is dedicated to the construction of the period homomorphism \(\text{per}_\omega: \pi_2 (G)\to z\). Section 6, which is the heart of the paper, contains the construction of a global group cocycle \(f:G\times G\to Z\) for simply connected groups \(G\) and any Lie algebra cocycle \(\omega\), where \(Z\) can be defined as \(z/ \Pi_\omega\). The so defined group \(Z\) is a Lie group iff \(\Pi_\omega\) is discrete, so that we obtain a Lie group extension iff \(\Pi_\omega\) is discrete. In Section 7 we eventually put all pieces together to prove the exactness of (1). The central result of Section 8 is Theorem 8.8 which gives a version of the exact sequence (1) for central Lie group extensions with smooth global sections. Section 9 is a collection of examples displaying various aspects in the description of the group \(\text{Ext}_{\text{Lie}} (G,Z)\) by the exact sequence (1)''.
scientific article; zbMATH DE number 1880701
scientific article