Exponentials of certain completions of the unitary form of a Kac-Moody algebra and associated groups
In the previous paper [J. Math. Kyoto Univ. 28, 633-659 (1988; Zbl 0683.17013)] the author considered certain classes of representations (\(\pi\),V) of a complex Kac-Moody algebra \({\mathfrak g}\) with a symmetrizable generalized Cartan matrix, and defined the spaces \(H_ m(\pi)\) of \(C^ m\)-vectors \((m=0,1,2,...,\infty,\omega)\) in completions of V. Let \({\mathfrak k}\) be the unitary form of \({\mathfrak g}\). Denote by \({\mathfrak k}_ m\) the closure of \({\mathfrak k}\) in the space \(H_ m(ad)\) of \(C^ m\)-vectors for the adjoint representation. In the paper under review the author proves the exponentiability of the elements in \({\mathfrak k}_{m+2}\) acting on \(H_ m(\pi)\) (Theorem 4.8), and the continuity of the exponential map (Theorem 5.2). For this purpose the author introduces the negative spaces \(H_{-m}(\pi)\) as the dual of \(H_ m(\pi)\) and defines the dual action \({\mathfrak k}_{m+1}\times H_{-m}(\pi)\to H_{-m-1}(\pi)\). For the action of \({\mathfrak k}_ m\) on the positive and negative spaces, he gives sharp estimates for the norm as linear maps \(H_ m(\pi)\to H_{m- 1}(\pi)\) and \(H_{-m+2}(\pi)\to H_{-m+1}(\pi)\). These estimates ensure that one can apply the criterion for exponentiability of an operator, which appears in Yosida's book on functional analysis.
scientific article; zbMATH DE number 4179614