Tame Fréchet structures for affine Kac-Moody groups. (English) Zbl 1311.22029

Freyn, Walter

Asian J. Math. 18, No. 5, 885-928 (2014).

The author uses functional analytic completions of affine Kac-Moody groups and algebras to construct a class of Lie groups and Lie algebras which admit manifold structures as tame Fréchet manifolds in the sense of Hamilton. The author also studies the adjoint action for these groups. One of the main results of the paper is the following:
Theorem. Groups of holomorphic loops and their real forms are tame Fréchet Lie groups.
Furthermore, it is shown that, for the loop groups of semisimple Lie algebras, the exponential map is not a local diffeomorphism. The paper also extends the theory of polar actions from the classical Hilbert space setting to the setting of tame Fréchet spaces. Finally, it is shown that all complex forms, real forms and quotient spaces of affine Kac-Moody groups that are necessary for the construction of affine Kac-Moody symmetric spaces are tame Fréchet manifolds.

Reviewer: Volodymyr Mazorchuk (Uppsala)

MSC:

22E67	Loop groups and related constructions, group-theoretic treatment
20G44	Kac-Moody groups

Keywords:

loop group; loop algebra; Kac-Moody group; affine algebra; completion; tame Fréchet space