# Tag Info

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 ...
• 43
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, ...
• 102k

### 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 ...
• 4,786
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 ...
• 102k
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 $p$...

### 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 ...
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 ...
• 1,313
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 ...
• 432k
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 ...
• 102k
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(\...
• 4,809
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, &...