### A Cartan decomposition for p-adic loop groups.

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We show that a Poisson Lie group (G,π) is coboundary if and only if the natural action of G×G on M=G is a Poisson action for an appropriate Poisson structure on M (the structure turns out to be the well known ${\pi}_{+}$). We analyze the same condition in the context of Hopf algebras. A quantum analogue of the ${\pi}_{+}$ structure on SU(N) is described in terms of generators and relations as an example.

Let A be a locally convex, unital topological algebra whose group of units ${A}^{\times}$ is open and such that inversion $\iota :{A}^{\times}\to {A}^{\times}$ is continuous. Then inversion is analytic, and thus ${A}^{\times}$ is an analytic Lie group. We show that if A is sequentially complete (or, more generally, Mackey complete), then ${A}^{\times}$ has a locally diffeomorphic exponential function and multiplication is given locally by the Baker-Campbell-Hausdorff series. In contrast, for suitable non-Mackey complete A, the unit group ${A}^{\times}$ is an analytic Lie group without...

The study of diffeomorphism group actions requires methods of infinite dimensional analysis. Really convenient tools can be found in the Frölicher - Kriegl - Michor differentiation theory and its geometrical aspects. In terms of it we develop the calculus of various types of one parameter diffeomorphism groups in infinite dimensional spaces with smooth structure. Some spectral properties of the derivative of exponential mapping for manifolds are given.

The main result of the present paper is an exact sequence which describes the group of central extensions of a connected infinite-dimensional Lie group $G$ by an abelian group $Z$ whose identity component is a quotient of a vector space by a discrete subgroup. A major point of this result is that it is not restricted to smoothly paracompact groups and hence applies in particular to all Banach- and Fréchet-Lie groups. The exact sequence encodes in particular precise obstructions for a given Lie algebra...