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1 vote

Applying an $SO(3)$ geodesic onto a unit vector results in a circle on a sphere

For the converse, there is a quick geometric construction. Consider the plane containing the circle and any vector $\omega=[\omega_x,\omega_y,\omega_z]^T$ normal to the plane. Pick any $b_0$ on the ...
pwensing's user avatar
1 vote

Alternative between 2-step solvable or perfect

Here are two observations to help you to classify such Lie groups: If $G$ is a noncompact connected semisimple Lie group, it contains a closed subgroup locally isomorphic to $SL(2,\mathbb R)$ and, ...
Moishe Kohan's user avatar
3 votes

uniqueness of generators of Lie groups

Qiaochu has already answered this in the comments but I wanted to put together a longer answer as I see this confusion fairly regularly here. It is quite common in Physics to use "the" a bit ...
Callum's user avatar
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1 vote
Accepted

The action of $SL(n,\Bbb{R})$ on the tangent space of $SL(n,\Bbb{R})/SO(n)$.

Since you asked about the case $n=2$: If such a representation exists, it is a nontrivial linear (real) 2-dimensional representation $\rho$. Moreover, since the center of $SL(2,\mathbb R)$ acts ...
Moishe Kohan's user avatar
1 vote

The action of $SL(n,\Bbb{R})$ on the tangent space of $SL(n,\Bbb{R})/SO(n)$.

You are making certain (very interesting !) mistakes. This is not quite my field, so I apologize if I say something imprecise. The overall issue is the question of "the tangent space of $P$ at $p$...
Fançois Gatine's user avatar
0 votes

Can somebody explain the plate trick to me?

Just a comment: As you point out, the video currently on the “Plate Trick” Wikipedia page doesn’t show the plate trick but rather the belt trick. In the plate trick, the connection to the plate (or ...
Mark Hunter's user avatar
1 vote

Motivation for Studying Nilpotent and Solvable Lie Algebras

I know it's been over 7 years, but I'll post this excerpt from the book "Emergence of the theory of Lie groups" by Thomas Hawkins because I find the accepted answer quite dismissive of the ...
Alex Bogatskiy's user avatar
3 votes
Accepted

Irreps of $SU(3)/\mathbb{Z}_3$ from irreps of $SU(3)$

(All representations are complex and finite-dimensional throughout except for at a handful of points where I talk about real representations.) In general, if $V$ is an irreducible representation of a ...
Qiaochu Yuan's user avatar
3 votes
Accepted

Isometry group of hyperbolic space

To get this off the "unanswered list." Canary is simply wrong when he says that $O_o(n,1)$ is the isometry group of the hyperbolic space. The group $O_o(n,1)$ is just the (unique) index 2 ...
Moishe Kohan's user avatar
3 votes
Accepted

Why $\omega|_S\in\Omega^1(S,\mathfrak{g})$ iff $\omega(\nabla,e)\in\Omega^1(U,\mathfrak{g})$?

The connection $\omega\in\Omega^1(\text{Fr}(TM),\mathfrak{gl}_r)$ induced by $\nabla$ is characterized by the following property: For each local frame $e:U\to\mathrm{Fr}(TM)$ $$ e^*\omega=\omega(\...
Claire's user avatar
  • 4,809
0 votes

Finding inverse to inner automorphism (Humphreys' Lie Algebra book)

Just to take this off the "unanswered" list, Qiaochu Yuan writes in a comment, "It's just a geometric series. But also you can just take $\exp(-\delta)$." and OP raynea confirms, &...
Torsten Schoeneberg's user avatar
4 votes
Accepted

Does the definition of Lie groups rely on invariance of domain?

The definition of a Lie group makes no reference to dimension, or tangent spaces, or invariance of domain. Read it carefully. All you need is a smooth manifold $G$ and a smooth map $m : G \times G \to ...
Qiaochu Yuan's user avatar
1 vote
Accepted

Conjugacy action of $SO(2m)$ on $O(2m)/U(m)$

As I suggested in a hint, it is useful to consider first the case $n=2$. Then there are exactly two order 4 rotations in the plane, namely, $J_1$ and $-J_1$ (rotations by $\pi/2$ and $-\pi/2$). They ...
Moishe Kohan's user avatar
0 votes
Accepted

Prove that $XY-YX\in \mathfrak{g}$ if $X,Y\in \mathfrak{g}$, where $\mathfrak{g}$ is the Lie algebra of a matrix Lie group $G$.

The answer that you have linked from José Carlos Santos uses that, if $V$ is a finite-dimensional subspace of $\mathbb{C}^n$, then $V$ is closed in $\mathbb{C}^n$ because (as mentioned in the second ...
tiral's user avatar
  • 103
1 vote
Accepted

Geometric interpretation and upper bounds on Euclidean distance in Lie algebra of SO(3)

The 2-norm on the Lie algebra of $SO(3)$ (the space of skew-symmetric $3\times 3$ matrices) is, up to a constant multiple, the Frobenius norm, i.e. with this norm $SO(3)$ is a Riemannian manifold. I ...
Moishe Kohan's user avatar
2 votes
Accepted

Question on the Proof of Proposition 6.6 in the Book Introduction to Lie Algebras by Karin Erdmann and Mark J. Wildon

From Lemma 5.5, I can see that if $z \in L$, then $a(zv) = z(av) = \lambda(a)zv$, thus $zv$ is also a eigenvector for $a$, where $a \in A$. I don't see the connection of it with eigenvector of $z$ in $...
Matthew Towers's user avatar

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