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Search: MSC category 22E67 ( Loop groups and related constructions, group-theoretic treatment [See also 58D05] )

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1. CJM Online first

Glöckner, Helge
Completeness of infinite-dimensional Lie groups in their left uniformity
We prove completeness for the main examples of infinite-dimensional Lie groups and some related topological groups. Consider a sequence $G_1\subseteq G_2\subseteq\cdots$ of topological groups~$G_n$ such that~$G_n$ is a subgroup of $G_{n+1}$ and the latter induces the given topology on~$G_n$, for each $n\in\mathbb{N}$. Let $G$ be the direct limit of the sequence in the category of topological groups. We show that $G$ induces the given topology on each~$G_n$ whenever $\bigcup_{n\in \mathbb{N}}V_1V_2\cdots V_n$ is an identity neighbourhood in~$G$ for all identity neighbourhoods $V_n\subseteq G_n$. If, moreover, each $G_n$ is complete, then~$G$ is complete. We also show that the weak direct product $\bigoplus_{j\in J}G_j$ is complete for each family $(G_j)_{j\in J}$ of complete Lie groups~$G_j$. As a consequence, every strict direct limit $G=\bigcup_{n\in \mathbb{N}}G_n$ of finite-dimensional Lie groups is complete, as well as the diffeomorphism group $\operatorname{Diff}_c(M)$ of a paracompact finite-dimensional smooth manifold~$M$ and the test function group $C^k_c(M,H)$, for each $k\in\mathbb{N}_0\cup\{\infty\}$ and complete Lie group~$H$ modelled on a complete locally convex space.

Keywords:infinite-dimensional Lie group, left uniform structure, completeness
Categories:22E65, 22A05, 22E67, 46A13, 46M40, 58D05

2. CJM 2006 (vol 58 pp. 625)

Mohrdieck, Stephan
A Steinberg Cross Section for Non-Connected Affine Kac--Moody Groups
In this paper we generalise the concept of a Steinberg cross section to non-connected affine Kac--Moody groups. This Steinberg cross section is a section to the restriction of the adjoint quotient map to a given exterior connected component of the affine Kac--Moody group. (The adjoint quotient is only defined on a certain submonoid of the entire group, however, the intersection of this submonoid with each connected component is non-void.) The image of the Steinberg cross section consists of a ``twisted Coxeter cell'', a transversal slice to a twisted Coxeter element. A crucial point in the proof of the main result is that the image of the cross section can be endowed with a $\Cst$-action.


3. CJM 2003 (vol 55 pp. 969)

Glöckner, Helge
Lie Groups of Measurable Mappings
We describe new construction principles for infinite-dimensional Lie groups. In particular, given any measure space $(X,\Sigma,\mu)$ and (possibly infinite-dimensional) Lie group $G$, we construct a Lie group $L^\infty (X,G)$, which is a Fr\'echet-Lie group if $G$ is so. We also show that the weak direct product $\prod^*_{i\in I} G_i$ of an arbitrary family $(G_i)_{i\in I}$ of Lie groups can be made a Lie group, modelled on the locally convex direct sum $\bigoplus_{i\in I} L(G_i)$.

Categories:22E65, 46E40, 46E30, 22E67, 46T20, 46T25

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