All Questions
Tagged with lie-groups dg.differential-geometry
567 questions
145
votes
14
answers
50k
views
Why study Lie algebras?
I don't mean to be rude asking this question, I know that the theory of Lie groups and Lie algebras is a very deep one, very aesthetic and that has broad applications in various areas of mathematics ...
- 2,751
78
votes
7
answers
8k
views
Example of a manifold which is not a homogeneous space of any Lie group
Every manifold that I ever met in a differential geometry class was a homogeneous space: spheres, tori, Grassmannians, flag manifolds, Stiefel manifolds, etc. What is an example of a connected smooth ...
- 8,559
77
votes
7
answers
21k
views
What is the symbol of a differential operator?
I find Wikipedia's discussion of symbols of differential operators a bit impenetrable, and Google doesn't seem to turn up useful links, so I'm hoping someone can point me to a more pedantic discussion....
- 54.6k
53
votes
5
answers
8k
views
Beautiful descriptions of exceptional groups
I'm curious about the beautiful descriptions of exceptional simple complex Lie groups and algebras (and maybe their compact forms). By beautiful I mean: simple (not complicated - it means that we need ...
- 1,422
39
votes
3
answers
4k
views
Can every Lie group be realized as the full isometry group of a Riemannian manifold?
Suppose a finite-dimensional Lie group $G$ is given. Does there exist a connected manifold $M$ and a Riemannian metric $g$, such that $G$ is the full isometry group of $(M,g)$?
For example if I try to ...
- 2,830
35
votes
5
answers
4k
views
$G_2$ and Geometry
In a recent question Deane Yang mentioned the beautiful Riemannian geometry that comes up when looking at $G_2$. I am wondering if people could expand on the geometry related to the exceptional Lie ...
- 3,726
34
votes
1
answer
1k
views
Is there a geometric construction of hyperbolic Kac-Moody groups?
Just as the theory of finite-dimensional simple Lie algebras is connected to differential geometry and physics via the theory of simple Lie groups, the theory of affine Lie algebras was connected to ...
- 22.3k
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
33
votes
1
answer
4k
views
Isometry group of a homogeneous space
Background
Let $(M,g)$ be a finite-dimensional riemannian (or more generally pseudoriemannian) manifold. Suppose that I know that a certain Lie group $G$ acts transitively and isometrically on $M$ ...
- 33.3k
31
votes
3
answers
3k
views
Rep Theory Consequences of Bott--Weil--Borel
I've been getting interested in the (Bott--)Borel--Weil theorem lately. As a (mainly) geometer it is very interesting to see representation appearing (from nowhere as far as I can see) in the theory ...
- 3,399
31
votes
3
answers
2k
views
What's the supersymmetric analogue of the Monster group?
Bosonic string theory lives in 26 dimensions, and it gives a conformal field theory where the field is a map from a Riemann surface to $\mathbb{R}^{24}$. The Leech lattice $L$ is an even unimodular ...
- 22.3k
28
votes
2
answers
2k
views
What is special to dimension 8?
Dimension $8$ seems special, as the partial list below might indicate.
Is there any overarching reason that dim-$8$ is "more special" than, say, dim-$9$?
Surely it isn't it, in the end, simply because ...
27
votes
10
answers
2k
views
Examples of Kan extensions, adjunctions, and (co)monads in analysis, Lie theory, and differential geometry?
In introductory texts on category theory, it seems like the majority of examples come from algebraic topology, algebra, and logic.
What are some good examples of Kan extensions, adjunctions, and (co)...
27
votes
2
answers
2k
views
Intuition for symplectic groups
My question essentially breaks down to
How do you, a working mathematician, think about (real) symplectic groups? How do you visualize symplectic (linear) transformations? What intuition do you ...
- 779
25
votes
1
answer
702
views
The de Rham complex of the octonionic projective spaces
The complex projective space $\mathbb{CP}^n$ is a complex manifold, and hence its de Rham complex carries a representation of the complex numbers in the form of its complex structure. The quaternionic ...
- 835
24
votes
2
answers
2k
views
Is it possible to realize the Moebius strip as a linear group orbit?
On MSE this got 5 upvotes but no answers not even a comment so I figured it was time to cross-post it on MO:
Is the Moebius strip a linear group orbit? In other words:
Does there exists a Lie group $ ...
- 4,081
24
votes
5
answers
2k
views
Lie groups vs Lie monoids
Does there exist a well developed theory of a class of objects which might rightfully be called Lie monoids? By this I mean with axioms similar to those of Lie groups, but with the axiomatic existence ...
- 2,099
22
votes
6
answers
3k
views
Automorphism group of real orthogonal Lie groups
I would like to understand what is the "outer-automorphism group" $Out$ of $SO(p,q)$ and $O(p,q)$, where $p+q >0$ and $pq \neq 0$. My working definition of $Out$ is as follows:
Let us denote by $...
- 2,816
22
votes
1
answer
720
views
Does $E_8$ know $Spin(7)$?
One way to define the compact group $Spin(7)$ is as the stabilizer of a certain 4-form on Euclidean $\mathbb R^8$ (see e.g. this MO question). This 4-form can be defined in various ways. For example,...
- 54.6k
20
votes
3
answers
9k
views
Curvature of a Lie group
Since a lie group is a manifold with the structure of a continuous group, then each point of the manifold [Edit: provided we fix a metric, for example an invariant or bi-invariant one] has some scalar ...
- 251
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
20
votes
3
answers
2k
views
Integral cohomology of $SU(n)$ - looking for constants
I am interested in explicit generators of the cohomology $H^\bullet(SU(n),\mathbb{Z})$. Let $\omega = g^{-1} dg$ be the Maurer-Cartan form on $SU(n)$. The forms $\alpha_3,\alpha_5,\dots,\alpha_{2n-1}$,...
- 653
20
votes
4
answers
3k
views
homotopy type of connected Lie groups
Is there a simple proof (short and low-tech) of the following fact:
(E. Cartan) A connected real Lie group $G$ is diffeomorphic (as a manifold) to
$K\times\mathbb{R}^n$ where $K$ is a maximal ...
- 7,609
20
votes
3
answers
2k
views
The invariant 3-form on a compact Lie group
Let $G$ be a compact Lie group. We have the well-known Maurer-Cartan left-invariant and right-invariant $1$-forms $\theta$ and $\bar\theta$ in $\Omega^1(G, \mathfrak{g})$, probably discussed in every ...
- 1,525
18
votes
1
answer
2k
views
The group of isometries of a manifold is a Lie group, isn't it?
Let $M$ be a connected finite dimensional topological manifold and $g$ be any metric on it that induces the topology of $M$ ($g$ is not a Riemannian metric). How to prove that the group of isometries ...
- 14.3k
18
votes
1
answer
1k
views
How come Cartan did not notice the close relationship between symmetric spaces and isoparametric hypersurfaces?
Elie Cartan made fundamental contributions to the theory of Lie groups and their geometrical applications. Among those, we can list the introduction of the remarkable family of Riemannian symmetric ...
- 4,729
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
17
votes
3
answers
5k
views
one-parameter subgroup and geodesics on Lie group
Hi,
Given a Matrix Lie Group, I would like to know if the one-parameter subgroups (which can be written as $\exp^{tX}$) are the same as the geodesics (locally distance minimizing curves). Geodesics ...
- 173
16
votes
1
answer
2k
views
A careful roadtrip from locally symmetric spaces to algebra
I'm trying to break the classification of locally riemannian symmetric spaces to little steps to make it more comprehensible (and s.t. the technical details can be verified without drowning completely)...
- 7,789
16
votes
2
answers
756
views
Can a sphere glued into a soft 3d-mattress rotate continuously? (manifolds, SU(2) and the belt trick)
The question is triggered by the wonderful animations by Jason Hise:
https://www.youtube.com/watch?v=LLw3BaliDUQ
https://www.youtube.com/watch?v=6Ul_-ABYaYU
https://www.youtube.com/watch?v=...
16
votes
1
answer
2k
views
Construction of the Lie functor: left vs. right invariant vector fields on Lie groups and Lie groupoids
When constructing the Lie algebra $L(G)$ of a Lie group $G$, one usually uses the identification of the tangent space $T_1 G$ with left invariant vector fields $\mathcal{V}^l(G)$ to construct the Lie ...
- 1,040
15
votes
8
answers
8k
views
Lie Groups and Manifolds
I'm trying to get a better handle on the relation between Lie groups and the Manifolds they correspond to. Firstly, is the relationship injective? that is, does each Lie group correspond to a unique ...
- 445
15
votes
3
answers
1k
views
Is SO(2n+1)/U(n) a symmetric space?
I am a physics student with only a rudimentary knowledge of differential geometry, so please feel free to point out if I miss something elementary / trivial.
According to https://arxiv.org/abs/1408....
- 305
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
3
answers
1k
views
Explicit formulas for Carnot-Carathéodory distances on Carnot groups
Let $G$ be a Carnot group (aka stratified group), so that $G$ is a connected and simply connected finite-dimensonal Lie group, whose Lie algebra $\mathfrak{g}$ admits a decomposition $\mathfrak{g} = ...
- 29.8k
15
votes
1
answer
2k
views
Intuition for the Cartan connection and "rolling without slipping" in Cartan geometry
Consider a Cartan geometry $\pi: \mathcal{G} \to M$ with Cartan connection $\omega$ modelled on the Klein geometry $(G, H)$.
The Cartan connection is supposed to formalize what it means to "roll ...
- 6,025
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
15
votes
0
answers
637
views
"Homogeneity" of the Hopf fibration $S^7\to S^{15}\to S^8$ [closed]
My question has to do with an apparent contradiction I get regarding the Hopf fibration $S^7\to S^{15}\to S^8$. Namely, the two following statements cannot be true at the same time (but I do not see ...
- 5,891
15
votes
0
answers
927
views
How does duality of symmetric spaces explain the hyperbolic cosine theorem?
There is a well-known duality between compact symmetric spaces and symmetric spaces of noncompact type. Basically it goes as follows: If $$G/K$$ is a symmetric space of noncompact type, $$g=k+p$$ the ...
- 10.4k
14
votes
4
answers
8k
views
Some basic questions about Chern-Simons theory
Let the Chern-Simons lagrangian for a group $G$ be,
$$L= k \epsilon^{\mu \nu \rho} Tr[A_\mu \partial _ \nu A_\rho + \frac{2}{3} A_\mu A_\nu A_\rho]$$
Then it is claimed that on "infinitesimal" ...
- 3,541
14
votes
2
answers
2k
views
If compact connected Lie groups are homeomorphic as topological space, are they isomorphic as Lie groups?
Let $G_{1}$ and $G_{2}$ be compact connected Lie groups.
If $G_{1}$ and $G_{2}$ are homeomorphic as topological spaces, are
they isomorphic as Lie groups?
- 491
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
14
votes
3
answers
982
views
Is the space of volume-preserving maps path-connected?
This is a clarification of another post of mine.
Fix $n$ a positive integer. Let $SL(n)$ have its usual matrix representation, so that it really is the codimension-one subset of $M(n) = \mathbb R^{n^...
- 54.6k
14
votes
2
answers
1k
views
When is a homogeneous space a variety?
Let $G$ be a Lie group and let $H$ be a closed subgroup of $G$. Then $G/H$ may not be a group, but it will be a homogeneous space for $G$ with stabilizers conjugate to $H$. Sometimes, this is a ...
- 16k
14
votes
1
answer
2k
views
what is the universal cover of GL(2,R)?
In the theory of Bridgeland stability conditions one has an action of the universal cover $G'$ of $G = GL^+(2,\mathbb R)$.
What is G'?
I know there is concrete description in terms of pairs (M,f) ...
- 303
14
votes
1
answer
681
views
If an equivariant map is smooth on diagonal matrices, is it smooth everywhere?
This is a followup from a question I asked on math.SE, which received a helpful answer but unfortunately not a complete one. $\def\Sym{\mathrm{Sym}_{n\times n}}$
$\def\s{\mathrm{Sym}}\def\sp{\s^+}$Let ...
13
votes
4
answers
4k
views
Classification of $SU(2)$ principal fibre bundles over four-dimensional manifolds
I would like to find a pedagogical reference where the classification, up to isomorphism, of principal $SU(2)$ bundles over a four-dimensional compact, oriented manifold is explained. In particular I ...
- 2,816
13
votes
2
answers
3k
views
What's the Lipschitz constant of the exponential map for $\mathrm{SO}(n,R)$?
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\so{\mathfrak{so}}$Consider the Lie algebra $\so(n)$ equipped with the metric $\langle e_i \wedge e_j, e_k \wedge e_l \rangle = \delta_{i,k} \delta_{j,l}...
- 4,466
13
votes
2
answers
9k
views
Maurer-Cartan form
I suppose given a Lie Group ($G$) and its corresponding Lie Algebra ($\mathfrak{g}$) every element in its dual defines a Maurer-Cartan form on the whole Lie Group?
Let $\omega \in \mathfrak{g}^*$ be ...
- 3,541
13
votes
2
answers
1k
views
What is known about Lie groups with (strictly) positive curvature?
If we consider $G$ a compact Lie group, there is a left invariant Riemannian metric whose the sectional curvature is nonnegative (see Milnors' paper). When can we find a left invariant metric that has ...
- 245