For example, the symmetries of a topological space are homeomorphisms. When the transformations of a set form a group, we call the group a transformation group and the set a G-set.

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We also say that the group G acts on the set X. The study of these groups will reveal a lot about the set itself.

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In this thesis we are interested in topological transformation groups i. We will, in particular, study the case where the topological space X is completely regular and where the transformation group G is a Lie group.

## Homori infinite dimensional lie transformation groups

The theory of Lie groups is vast and they have a well-understood structure. Our main goal is to present and prove the so-called slice theorem which is one of the most important results in the theory of transformation groups. A slice in a G-space X characterizes the action of G locally in an invariant neighbourhood of an orbit of X. The slice theorem i. In order to prove the existence of slices in the case of non-compact Lie groups, the way that G acts needs to somehow be restricted.

It turns out that proper action is the right way to do this.

The existence of slices for proper actions of noncompact Lie groups was firsst proved by Palais in We will present these two proofs in detail and compare them. It seems that you're in Germany. We have a dedicated site for Germany.

This collection of invited expository articles focuses on recent developments and trends in infinite-dimensional Lie theory, which has become one of the core areas of modern mathematics. The book is divided into three parts: infinite-dimensional Lie super- algebras, geometry of infinite-dimensional Lie transformation groups, and representation theory of infinite-dimensional Lie groups. Part A is mainly concerned with the structure and representation theory of infinite-dimensional Lie algebras and contains articles on the structure of direct-limit Lie algebras, extended affine Lie algebras and loop algebras, as well as representations of loop algebras and Kac—Moody superalgebras.

The articles in Part B examine connections between infinite-dimensional Lie theory and geometry. The topics range from infinite-dimensional groups acting on fiber bundles, corresponding characteristic classes and gerbes, to Jordan-theoretic geometries and new results on direct-limit groups.

## Developments and Trends in Infinite-Dimensional Lie Theory

The analytic representation theory of infinite-dimensional Lie groups is still very much underdeveloped. The articles in Part C develop new, promising methods based on heat kernels, multiplicity freeness, Banach—Lie—Poisson spaces, and infinite-dimensional generalizations of reductive Lie groups.

Contributors: B. Allison, D. Bertram, J. Faulkner, Ph. Gille, H.

## Infinite dimensional lie transformations groups

Neeb, E. Neher, I. Penkov, A. Pianzola, D.