Groups associated with unitary forms of Kac-Moody algebras
The group associated with Kac-Moody algebras and its unitary forms were constructed and studied by Kac-Peterson and others [cf. \textit{V. G. Kac}, Infinite dimensional Lie algebras (2nd edition, Cambridge University Press 1985; Zbl 0574.17010); Astérisque, No.Hors Ser. 1985, 165-208 (1985; Zbl 0625.22014)]. The group is constructed by the exponential map defined on the real root vectors and there do not exist the one-parameter subgroups corresponding to imaginary roots. Now, let \({\mathfrak k}\) be the unitary form of a complex Kac-Moody algebra \({\mathfrak g}\) with symmetrizable generalized Cartan matrix. In this paper, the author constructs the group associated to \({\mathfrak k}\) in such a way that the exponential map is defined for every element of the algebra \({\mathfrak k}.\) Namely, let \(\Lambda\) be a dominant integral element in the dual of the Cartan subalgebra of \({\mathfrak g}\), L(\(\Lambda)\) the irreducible highest weight module for \({\mathfrak g}\) with highest weight \(\Lambda\). Then, L(\(\Lambda)\) has a canonical pre-Hilbert space structure under which each element of \({\mathfrak k}\) acts on L(\(\Lambda)\) as an anti-symmetric operator. The author defines the exponential map from \({\mathfrak k}\) into the group U(\(\Lambda)\) of all unitary operators on the completion H(\(\Lambda)\) of L(\(\Lambda)\) equipped with the strong operator topology. The group associated with \({\mathfrak k}\) is defined to be the closed subgroup of U(\(\Lambda)\) generated by exp \({\mathfrak k}\). The author discusses the relation between the constructed group and the groups constructed on the lowest weight modules in the same manner and also the relation between these groups and the groups treated by Kac-Peterson.
scientific article; zbMATH DE number 4061460