2
votes
0answers
33 views

Inner automorphisms of Lie groups

I have a few questions about $Aut(G)$, when $G$ is a Lie group. It was proven by Hochschild that if $G/G_0$ is finitely generated, then $Aut(G)$ is a Lie group with at most countably many components. ...
0
votes
0answers
35 views

looking for books on topological semigroup:

I'm looking for several books on topological semigroup: Topological semigroups: history, theory, applications. Karl Heinrich Hofmann Mathematics Research Library, Tulane University. The ...
2
votes
3answers
145 views

What is an awesome book as an introduction to hyper groups

I'm a grad studen and i'm choosing an area to follow on my doctorate (in?) and I've been thinking about extension of topological group theory results to topological hypergroups, but for that i need to ...
2
votes
1answer
24 views

Is every regular paratopological group completely regular?

This problem is presented as an open problem 1.31. on p.26 of Arhangel'skii-Tkachenko, Topological groups and related structures. Is this problem still open?
2
votes
0answers
34 views

Analytic/Smooth/Continuous maps between a manifold and itself

Let us suppose that $M_{\omega}$ is a connected real-analytic manifold of dimension $n$. Then there is an associated smooth structure, $\mathcal{C}^r$ structure ($r$ non-negative integer) on it. Let ...
2
votes
1answer
42 views

Familiarizing with the Grothendieck topos $\mathbf{B}G$.

I am trying to familiarize with the Grothendieck topos $\mathbf{B}G$ of continuous $G$-sets, where $G$ is a topological group. I am unfortunately not very familiar with working with different ...
3
votes
0answers
49 views

Is Steinhaus theorem ever used in topological groups?

Steinhaus theorem in $\mathbb{R}^d$ says that for $E\subset\mathbb{R}^d$ with positive measure, $E-E:=\{x-y:x,y\in E\}$ contains an open neighborhood of the origin. And for locally compact Hausdorff ...
10
votes
1answer
106 views

Good book for studying $S_\infty$.

I'm looking for any books with some good information involving $S_\infty$ and other Polish groups. Specifically interested in $S_\infty$. This is an extremely amazing topological group, now having ...
4
votes
1answer
56 views

reference request: proof that group characters are a basis for $L^2$

I know the following must be very standard, but I haven't found it in any of the functional analysis books to which I have access. Do you know where I can find a self-contained proof?: If $G$ is a ...
1
vote
0answers
40 views

The structure of locally compact abelian groups by D. L. Armacost.

I do not have access to this important book: The structure of locally compact abelian groups by D. L. Armacost. I need to the Example 6.4 of this book. I will be grateful if you can help
11
votes
1answer
140 views

Conditions for a topological group to be a Lie group.

In flipping through the Springer lecture notes on Serre's 1964 'Lie Algebras and Lie Groups' lectures at Harvard, I found this pair of suprising results (page 157): Let $G$ be a locally compact ...
3
votes
1answer
66 views

Representation of topological groups

I am looking for a good book of topological representation. I have a very good insight of representation theory of finite groups, and I want to explore topological representations. I saw a book by ...
3
votes
1answer
106 views

A short exact sequence of groups and their classifying spaces

Suppose that we have a short exact sequence of topological groups: $$1 \to H \to G \to K \to 1.$$ I have found some papers mentioning that the above sequence induces a fibration: $$BH \to BG \to BK.$$ ...
7
votes
2answers
204 views

Good book on topological group theory?

I'm looking for a good introduction to the theory of locally compact groups and their representations. It may assume the reader to be familiar with basic group theory, topology and measure theory.
0
votes
0answers
55 views

Is the Hilbert-Smith conjecture still unsolved?

Conjecture Let $G$ be a locally compact topological group. If $G$ has a continuous faithful group action on an $n$-manifold, then $G$ is a Lie group. Is this conjecture still unsolved? Is ...
1
vote
0answers
35 views

History: continuously differentiable groups over the real numbers

Continuously differentiable groups over the real numbers are all isomorphic to addition, as is well-known, but who proved it and when?
3
votes
1answer
153 views

Research Sources for $SL(2,R)$

Can anyone guide me to a good site for the special linear group $SL(2,R)$, especially one that goes deep into its subgroup and normal subgroup? Book recommendations would be great too.
11
votes
1answer
195 views

subgroup of connected locally compact group

I need a reference or a short proof for the following property: A nontrivial connected locally compact group $G$ contains an infinite abelian subgroup.
0
votes
1answer
87 views

existence of infinite abelian subgroup in infinite locally compact groups

1) Let $G$ be an infinite locally compact group. Does there exist an infinite abelian locally compact subgroup of $G$? Rem: I know that there exists an infinite abelian subgroup in every infinite ...
1
vote
1answer
80 views

Centralizers in reductive Liegroups = unimodular?

Let $G$ be a real reductive group. Why is the centralizer of an element unimodular? What is a reference?
8
votes
2answers
352 views

Exact sequence in a nonabelian category [previously: “Exact sequence for topological groups?”]

If $A$, $B$, and $C$ are topological groups, and $f: A \to B$ and $g: B \to C$ are two continuous group homomorphisms, what does it usually mean for $$1 \to A \stackrel{f}{\to} B \stackrel{g}{\to} C ...
4
votes
1answer
303 views

Completion of Topological Group with Metric

Related to this question, I'm having trouble understanding the construction of the completion of a topological group with metric structure. In particular, under what conditions is the completion also ...