All Questions
36 questions
33
votes
8
answers
9k
views
"Modern" proof for the Baker-Campbell-Hausdorff formula
Does someone has a reference to a modern proof of the Baker-Campbell-Hausdorff formula?
All proofs I have ever seen are related only to matrix Lie groups / Lie algebras and
are not at all geometric (...
- 2,074
20
votes
4
answers
3k
views
Online References for Cartan Geometry
I would like to learn more about Cartan Geometry ("les espaces généralisés de Cartan"). I ordered Rick Sharpe's book "Differential Geometry: Cartan's generalization...", which would take a long time ...
- 5,215
17
votes
2
answers
2k
views
Where did Sophus Lie write the group commutator for two one parameter groups
If $X,Y$ are vector fields and $\def\Fl{\operatorname{Fl}}\Fl^X_t$ and $\Fl^Y_t$ their local flows, let $[\Fl^X_t,\Fl^Y_t]:= \Fl^Y_{-t}\Fl^X_{-t}\Fl^Y_t\Fl^X_t$ denote the group commutator of the ...
- 25.3k
15
votes
1
answer
612
views
Is the subgroup $\mathrm{Diff}(M,S)$ of $\mathrm{Diff}(M)$ a Lie subgroup?
Denote by $\mathrm{Diff}(M)$ the Lie group of smooth diffeomorphisms on a compact smooth manifold. Its Lie algebra can be viewed as the Lie algebra $\mathfrak X(M)$ of vector fields on $M$. Now, given ...
- 2,789
15
votes
1
answer
951
views
Duistermaat and Kolk's lost chapters on Lie groups
In Duistermaat and Kolk's book Lie Groups, it is written in the preface that "the text contains references to chapters belonging to a future volume". I could not find this second volume anywhere. Has ...
- 151
14
votes
5
answers
1k
views
History of the notion of $(G,X)$-structure
I'm currently searching for sources and historical basis on the notion of $(G,X)$-structure as it appears in Thurston's work.
So far, it appears that he was the first to set it. Many mathematicans ...
- 403
10
votes
3
answers
2k
views
nth term in the Baker-Campbell-Hausdorff formula
I am trying to prove a result for which I need the nth term of the Baker-Campbell-Hausdorff formula. I came at this particular result (which is not of significance for the question, but mentioning for ...
- 245
9
votes
1
answer
874
views
Proofs that the conformal group in dimension $\ge 3$ is a Lie group
Let $M$ be a smooth manifold of dimension $\ge 3$, equipped with a conformal structure (or a Riemannian metric). Then, the group of conformal diffeomorphisms is a finite dimensional Lie group.
A ...
- 6,741
9
votes
2
answers
639
views
Contact distributions on $(G_2,P)$-type Cartan geometries in dimension 5
Up to topology, the 5D homogeneous space
$$
G_2/P
$$
of the (real form of the) 14D exceptional Lie group $G_2$ is the 5D jet space
$$
M:=J^1(2,1)=\{(x,y,u,p,q)\}
$$
of scalar functions in two ...
- 2,527
8
votes
1
answer
633
views
Flag manifolds (=R-spaces): quotients by parabolic subgroups vs. isotropy representation
Real flag manifolds (also known as R-spaces) can be defined in two ways which I believe are equivalent although some fine print may have escaped me:
as a quotient of a semisimple real Lie group $G$ ...
- 32.5k
8
votes
1
answer
673
views
Classification of compact globally symmetric spaces
It is known that any connected compact Lie group $G$ is a finite quotient of the product of a compact simply connected semisimple Lie group $\tilde{G}$ and a torus $\mathbb{T}^n$ (see for example ...
- 193
6
votes
2
answers
1k
views
Parallel forms and cohomology of symmetric spaces
Let $G/H$ be a compact symmetric space. Then I believe the following is true: if $\alpha \in \Omega^k(G/H)$ and $\nabla$ the Levi-Civita connection, then
$$
(\alpha \text{ is induced by an $\...
- 3,244
5
votes
1
answer
978
views
Existence proof of Bourbaki, Differentiable and Analytic Manifolds
I am reading through Chapter III of Bourbaki, Lie Groups and Lie Algebras, and many proofs cite the Bourbaki volume Differentiable and Analytic Manifolds. I can't find this book anywhere. Does it ...
- 6,180
5
votes
1
answer
530
views
Geodesic distance on $\mathrm{SO}(n)$
$\DeclareMathOperator\SO{SO}$Recently I came across this old MSE post or this paper (w.o. proof) discussing the geodesic distance on $\SO(n)$ when it is equipped with the left-invariant Riemannian ...
- 364
5
votes
1
answer
147
views
Equivalence generated by Jacobian minors
Let $f,g:\mathbb{R}^m \to \mathbb{R}^n$ be two smooth functions and let $k$ be a strictly positive integer. Write $f \sim_k g$ if at each point in the domain, the determinants of all $k \times k$ ...
- 15.5k
5
votes
1
answer
134
views
Homogeneous representations of compact manifolds
There is a classification of effective transitive groups actions on the sphere by compact connected Lie groups, compare Besse "Einstein manifolds" 7.13 Examples.
Are there similar results ...
- 179
5
votes
0
answers
276
views
Fundamental group of compact globally symmetric spaces
The fundamental group of a globally symmetric space $M$ of compact type is known (see Loos [1], Borel [2]). The result can be formulated as follows: it is isomorphic to the quotient
$$(*) \quad \pi_1(...
- 1,123
4
votes
1
answer
295
views
On a result of Cartan for homogeneous manifolds arising from a quotient of discrete subgroups
I'm not sure if this is completely relevant to MO, let me know if this would be better on MSE.
I have been told today by a professor of mine that the following is a classic result of Cartan. Suppose $...
- 1,763
4
votes
2
answers
758
views
Riemannian metric of hyperbolic plane
I'm fishing for the origin of the idea to consider "trace scalar product" on the space of ($G$-)orthogonal projectors as means of defining a Riemannian metric on some subset of lines in a vector space....
- 8,597
4
votes
0
answers
552
views
Lattices of $\mathbb{R}^s \ltimes_\varphi \mathbb{R}^k$
Edit: Thoughts updated (22/3/2021).
I've come across with the following problem.
Let $G=\mathbb{R}^s \ltimes_\varphi \mathbb{R}^k$ where $\varphi:\mathbb{R}^s\to \mathrm{Aut}(\mathbb{R}^k)=\mathsf{GL}(...
4
votes
0
answers
367
views
Representation theory and associated bundles
I am looking for a text or set of notes that discusses the relationship between the representation theory of Lie groups and associated vector bundles, preferably using modern categorical language. For ...
- 6,025
3
votes
2
answers
351
views
Commutator 2-forms on Lie groups
Let $G$ be a compact Lie group and $\mathfrak g$ its Lie algebra.
For any $f$ in the dual space $\mathfrak g^*$, we can define a skew-symmetric bi-linear form on $\mathfrak g$ by $(A,B)\mapsto f([A,B])...
- 8,086
3
votes
2
answers
1k
views
Reference for homogeneous spaces
I am a graduate student of differential geometry.
I would like to get an overview over the way, how results are usually obtained for homogeneous spaces by Lie algebraic methods. By definition a ...
- 79
3
votes
1
answer
239
views
What is the curved version of the Tits fibration for $G_2$?
Let
$\require{AMScd}$
\begin{CD}
G_2/(P_1\cap P_2) @= G_2/(P_1\cap P_2)=:\mathbb{I}\\
@V \lambda V V @VV \pi V\\
\mathbb{Q}_5:=G_2/P_1 && G_2/P_2=:\mathbb{N}_5
\end{CD}
be the Tits ...
- 2,527
3
votes
3
answers
1k
views
Lie algebra bundle associated to a Lie group bundle
I was reading the paper Non abelian Differentiable gerbes (page 24) and came across notion of Lie algebra bundles associated to a Lie group bundle.
I am not comfortable with these notions and google ...
- 6,023
3
votes
0
answers
142
views
Conjugacy classes of Cartan subspaces in parahermitian symmetric spaces
Are there any good tables of the numbers of conjugacy classes of Cartan subspaces in pseudo-Riemannian symmetric spaces? Or a good method to count them? In particular, I am interested in the ...
- 954
3
votes
0
answers
129
views
Differential operators on a compact Lie group associated to bracket-generating sets
Let $G$ be a compact connected Lie group of dimension $m$ with Lie algebra $\mathfrak g$.
Let $\{X_1,\dots,X_h\}$ be a linearly independent set of $\mathfrak g$.
Assume that $\{X_1,\dots,X_h\}$ is ...
- 2,446
2
votes
1
answer
280
views
Unitary representation with fixed Casimir
Let $G$ be a connected reductive real Lie group with Lie algebra $\mathfrak{g}$. We denote by $\widehat{G}_u$ the unitary dual, that is the set of isomorphism classes of unitary reprensentation of $G$....
- 1,111
2
votes
1
answer
214
views
What is the space parametrising the curved sub-Cartan geometries of a flat Cartan geometry?
I'm basically wondering how to make "curved" the first column of the diagram
$\require{AMScd}$
\begin{CD}
P_1 @>\textrm{inclusion} >> G\\
@V \omega_0 V P_1\cap P_2 V @V\omega V P_2 V\\...
- 2,527
2
votes
1
answer
255
views
Parameter dependent differential equation in a Lie group
It is well-known that a linear differential equation in a finite-dimensional vector space depends continuously on some external parameters (for details see below). I search for an explicit reference ...
- 5,824
2
votes
2
answers
519
views
How to use that the Hessian is negative definite in this proof
Let $X$ be a Riemannian compact manifold on which acts a compact Lie group $H^+$. Let $f^+ : X \rightarrow \mathbb{R} $ be a smooth function on $X$. Consider a Lie subgroup $U$ of $H^+$ and suppose ...
- 139
2
votes
0
answers
115
views
Special class of bi-hamiltonian systems
A bi-Hamiltonian manifold is a manifold $M$ equipped with two compatible Poisson tensors $\pi_0$ and $\pi$.
I am interested in the case of a Lie group $G$ endowed with a multiplicatif Poisson tensor $...
- 513
1
vote
2
answers
341
views
Copies of ax+b inside the AN part of an Iwasawa decomposition?
As a relative novice to the structure theory of Lie algebras and Lie groups, the following is what I can gather from reading parts of Helgason's book DG, Lie groups and symmetric spaces and Knapp's ...
- 25.8k
1
vote
1
answer
157
views
Commuting time dependent vector fields and pullback invariance
Let $X_t, Y_t \in C^\infty(\mathbb{R}; \mathfrak{X}^\infty(M))$ be (smooth, or something else if it's necessary) time dependent vector fields.
Is there some analogue of the following fact in finite ...
- 193
1
vote
0
answers
132
views
About the classification of simply connected homogeneous 3-manifolds
I've read somewhere (but cannot locate the source) that the following classification holds: simply connected homogeneous 3-manifolds are either isometric to $S^2 \times \mathbb{R}$ or to a metric Lie ...
- 1,763
1
vote
0
answers
96
views
Largest dimensional Lie subgroup of $SU(N)$ [duplicate]
What is the largest (Lie) subgroup of $SU(n)$ in the sense of its dimension.
I am aware of this potential duplicate subgroup of SU(N) with maximal manifold dimension , however, the title of this ...
- 2,099