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Register a new userSolutions to open problems in Neebs recent survey on infinite-dimensional Lie groups
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Authors
Helge Glockner
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We solve three open problems concerning infinite-dimensional Lie groups posed in a recent survey article by K.-H. Neeb: (1) There exists a subgroup of some infinite-dimensional Lie group G which does not admit an initial Lie subgroup structure; (2) The pathology cannot occur if G is a direct limit of an ascending sequence of finite-dimensional Lie groups; (3) Every such direct limit group is a ``topological group with Lie algebra in the sense of Hofmann and Morris. Moreover, we prove a version of Borels Theorem announced in the survey, ensuring the existence of compactly supported smooth diffeomorphisms with given Taylor series around a fixed point p (provided the tangent map at p has positive determinant).
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Many infinite-dimensional Lie groups of interest can be expressed as a union of an ascending sequence of (finite- or infinite-dimensional) Lie groups. In this survey article, we compile general results concerning such ascending unions, describe the main classes of examples, and explain what the general theory tells us about these. In particular, we discuss: (1) Direct limit properties of ascending unions of Lie groups in the relevant categories; (2) Regularity in Milnors sense; (3) Homotopy groups of direct limit groups and of Lie groups containing a dense union of Lie groups; (4) Subgroups of direct limit groups; (5) Constructions of Lie group structures on ascending unions of Lie groups.
We generalize both the notion of polynomial functions on Lie groups and the notion of horizontally affine maps on Carnot groups. We fix a subset $S$ of the algebra $mathfrak g$ of left-invariant vector fields on a Lie group $mathbb G$ and we assume that $S$ Lie generates $mathfrak g$. We say that a function $f:mathbb Gto mathbb R$ (or more generally a distribution on $mathbb G$) is $S$-polynomial if for all $Xin S$ there exists $kin mathbb N$ such that the iterated derivative $X^k f$ is zero in the sense of distributions. First, we show that all $S$-polynomial functions (as well as distributions) are represented by analytic functions and, if the exponent $k$ in the previous definition is independent on $Xin S$, they form a finite-dimensional vector space. Second, if $mathbb G$ is connected and nilpotent we show that $S$-polynomial functions are polynomial functions in the sense of Leibman. The same result may not be true for non-nilpotent groups. Finally, we show that in connected nilpotent Lie groups, being polynomial in the sense of Leibman, being a polynomial in exponential chart, and the vanishing of mixed derivatives of some fixed degree along directions of $mathfrak g$ are equivalent notions.
We show that, in compact semisimple Lie groups and Lie algebras, any neighbourhood of the identity gets mapped, under the commutator map, to a neighbourhood of the identity.
In this paper we prove some properties of the nonabelian cohomology $H^1(A,G)$ of a group $A$ with coefficients in a connected Lie group $G$. When $A$ is finite, we show that for every $A$-submodule $K$ of $G$ which is a maximal compact subgroup of $G$, the canonical map $H^1(A,K)to H^1(A,G)$ is bijective. In this case we also show that $H^1(A,G)$ is always finite. When $A=ZZ$ and $G$ is compact, we show that for every maximal torus $T$ of the identity component $G_0^ZZ$ of the group of invariants $G^ZZ$, $H^1(ZZ,T)to H^1(ZZ,G)$ is surjective if and only if the $ZZ$-action on $G$ is 1-semisimple, which is also equivalent to that all fibers of $H^1(ZZ,T)to H^1(ZZ,G)$ are finite. When $A=Zn$, we show that $H^1(Zn,T)to H^1(Zn,G)$ is always surjective, where $T$ is a maximal compact torus of the identity component $G_0^{Zn}$ of $G^{Zn}$. When $A$ is cyclic, we also interpret some properties of $H^1(A,G)$ in terms of twisted conjugate actions of $G$.
This note is concerned with the geometric classification of connected Lie groups of dimension three or less, endowed with left-invariant Riemannian metrics. On the one hand, assembling results from the literature, we give a review of the complete classification of such groups up to quasi-isometries and we compare the quasi-isometric classification with the bi-Lipschitz classification. On the other hand, we study the problem whether two quasi-isometrically equivalent Lie groups may be made isometric if equipped with suitable left-invariant Riemannian metrics. We show that this is the case for three-dimensional simply connected groups, but it is not true in general for multiply connected groups. The counterexample also demonstrates that `may be made isometric is not a transitive relation.
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