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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
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
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
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
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
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
1 vote
Accepted

Question on the proof of Engel's Theorem step 1 in book Introduction to Lie Algebras

Those are two separate claims both of which they prove later. In fact they are simply stating what they plan to show in the proof. They show that $\tilde{A} = A \oplus \mathrm{Span}\{y\}$ is a larger ...
Callum's user avatar
  • 4,721
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

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