Loop groups and their actions on the corresponding completed affine Lie algebras
The paper under review is devoted to the study of groups of \(C^ k\)- loops in Lie groups. The author follows the method in the work of \textit{H. Garland} [Publ. Math., Inst. Haut. Étud. Sci. 52, 5-136 (1980; Zbl 0475.17004)]. Let \(L_ k\) be the space \(C^ k(S^ 1)\) and \(L_{k,\mathbb{R}}\) the subspace consisting of real functions (\(k=0,1,\dots,\infty\)). For a finite-dimensional manifold \(M\), let \(M(L_ k)\) denote the loop space of class \(C^ k\). In the first part of the paper the structure of \(M(L_ k)\) is studied. The main result here is: if \(M\) is a Lie group, then \(M(L_ k)\) can be provided with a canonical Lie group structure such that its tangent space is \((L_{k,\mathbb{R}})^{\dim M}\). Now let \(G\) be a finite-dimensional connected simply connected complex simple Lie group, \({\mathfrak g}\) the Lie algebra of \(G\). Denote \(\widetilde{G}_ k= G(L_ k)\), \(\widetilde {\mathfrak g}_ k={\mathfrak g}(L_ k)\). Let \(\widehat{\mathfrak g}_ k\) be a corresponding completed affine Lie algebra -- a one-dimensional central extension of \(\widetilde{\mathfrak g}_ k\). The group \(\widetilde{G}_ k\) acts in a natural way on \(\widetilde {\mathfrak g}_ k\) (by the adjoint representation). In the second part of the paper the author extends this action of \(\widetilde{G}_ k\) to an action on \(\widehat{\mathfrak g}_ k\) (with an explicit description). Moreover, he extends the action on \(\widehat {\mathfrak g}_ k\) to that on the so called extended affine Lie algebra \(\widehat{\mathfrak g}_ k^ e\). As an application explicit forms are given of the normalizers and centralizers in \(\widetilde{G}_ k\) of the Cartan subalgebras of \(\widehat {\mathfrak g}_ k\) and \(\widehat {\mathfrak g}_ k^ e\). It turns out that in both cases the corresponding quotient groups are canonically isomorphic to the usual affine Weyl group.
scientific article; zbMATH DE number 169032