Central extensions of current groups
Let \(G\) be the identity component of the current group \(C^\infty(M,K)\), where \(M\) is a compact manifold and \(K\) a finite- or infinite-dimensional Lie group. The authors consider continuous Lie algebra \(2\)-cocycles \(\omega\) on the Lie algebra \({\mathfrak g}=C^\infty(M,{\mathfrak k})\) of \(G\) which are of ``product type'', meaning that \(\omega(\xi,\eta):=[\kappa(\xi,d\eta)]\in {\mathfrak z} :=\Omega^1(M,Y)/ dC^\infty(M,Y)\) for some continuous, symmetric, invariant bilinear map \(\kappa : {\mathfrak k}\times {\mathfrak k}\to Y\) on the Lie algebra \({\mathfrak k}\) of \(K\), with values in a sequentially complete, locally convex space \(Y\). The article is centered around the question of whether there exists a central Lie group extension \(\widehat{G}\) of \(G\) corresponding to \(\omega\). It is known from the general extension theory developed by the second author [Ann. Inst. Fourier 52, 1365-1442 (2002; Zbl 1019.22012)] that \(\widehat{G}\) exists if and only if the period group \(\Pi_{M,\kappa}\subseteq {\mathfrak z}\) (the image of the period homomorphism \(\text{per}_\omega : \pi_2(G)\to {\mathfrak z}\)) is discrete and the adjoint action of \({\mathfrak g}\) on \({\mathfrak g}\oplus_\omega {\mathfrak z}\) integrates to a smooth representation of \(G\). The second condition is always satisfied in the present context; the Lie algebra action even integrates to a smooth representation of \(C^\infty(M,K)\) (Proposition III.3). As the authors show, for fixed \(\kappa\), the first condition is satisfied for every \(M\) if and only if it is satisfied for \(M={\mathbb S}^1\), i.e., in the case of loop groups (Theorem I.6). Hence \(\widehat{G}\) exists for general \(M\) if and only if it does so for \(M={\mathbb S}^1\). The proof is based on an explicit calculation of \(\Pi_{M,\kappa}\) in terms of \(\Pi_{{\mathbb S}^1,\kappa}\) which is quite remarkable, because \(\pi_2(C^\infty(M,K))\) is not well accessible when \(\dim(M)>2\). The authors then turn to finite-dimensional \(K\); in this case, a central extension \({\mathfrak z}/\Pi_{M,\kappa}\to \widehat{G}\to G\) is shown to exist for \(\kappa : {\mathfrak k}\times {\mathfrak k}\to V({\mathfrak k})\), the universal symmetric invariant bilinear map (Corollary III.7). If \(K\) is semisimple, this implies the existence of a central extension of \(G\) which is universal among all central extensions of \(G\) by abelian regular Fréchet-Lie groups (Theorem IV.5). So far, only a universal central extension of the universal covering group \(\widetilde{G}\) (rather than \(G\)) had been discussed in the literature [\textit{A. Pressley} and \textit{G. Segal}, Loop groups, Oxford University Press (1988; Zbl 0638.22009)], for \(K\) a simply connected, simple, compact Lie group. In a section which can be read independently of the others, the authors also study the problem to associate to a pair of automorphisms of Lie groups \(Z\) and \(G\) an automorphism of a given central extension \(\widehat{G}\) of \(G\) by \(Z\), and they describe sufficient conditions ensuring that actions of a Lie group \(R\) on \(Z\) and \(G\) by automorphisms give rise to a smooth automorphic action of \(R\) on \(\widehat{G}\).
scientific article; zbMATH DE number 1982575