13
votes

Accepted

### Maximum symmetry metric on $ \mathbb{C}P^n $

There's an easy counterexample to your guess: Let $M^6 = \mathrm{SU}(3)/\mathbb{T}^2$, where $\mathbb{T}^2\subset\mathrm{SU}(3)$ is the maximal torus (for example, the diagonal subgroup). In that ...

- 101k

5
votes

### Description of the generalized grassmannians and flag varieties (parabolic quotients) associated to the exceptional groups

Definitely not a full answer (and rather late to the party) but I am interested in the $E_7$ case myself so I have been doing a little computing of the dimensions of these flags that I thought was ...

- 747

5
votes

Accepted

### Injectivity of the cohomology map induced by some projection map

Ok, I will follow Fernando's advice and post an answer. I learned the computation below from the beginning of Pin(2)-equivariant Seiberg--Witten Floer homology and the triangulation conjecture.
The ...

- 8,728

5
votes

Accepted

### Question about maximal compact subgroups of Lie groups

$\DeclareMathOperator\Lie{Lie}\newcommand\g{\mathfrak g}\newcommand\C{{\mathbb C}}$$\g = \Lie(G)$ is maximal among subalgebras of $\g_\C = \Lie(G_\C)$ on which the Killing form is negative definite, ...

- 9,057

5
votes

### Maximum symmetry metric on $ \mathbb{C}P^n $

I just wanted to add two points:
A bi-invariant metric on a compact Lie group $G$ does not always induced the maximum symmetric metric on $G/H$. The most familiar examples are spheres: $S^{2n+1} = ...

- 6,011

4
votes

Accepted

### Generalization of $G/T \simeq G_\mathbb{C}/B$

No.
Take $G = SU(2)$, $G_{\mathbb C} = SL_2(\mathbb C)$, $G/B$ the complex projective line alias the sphere, $H$ the diagonal $U(1)$, $x$ any point other than the two fixed points of $H$, so that the ...

- 122k

4
votes

### Injectivity of the cohomology map induced by some projection map

OK, just noticed mme's comment, which considers the very same counterexample below with a clean one-line argument. I'm leaving this here in case someone finds something of any value, but I think mme's ...

- 14.3k

2
votes

Accepted

### Free $S^1$-action on compact homogeneous spaces

Here is a counterexample where $K$ is connected.
Let $W = SU_3/SO_3$ be the Wu manifold. This 5-dimensional manifold is simply-connected and has $H_2(W;\Bbb Z) = \Bbb Z/2$, and in particular, $\pi_2(W)...

- 8,728

2
votes

Accepted

### Question about coadjoint orbits of compact connected Lie groups

$\DeclareMathOperator{\Tr}{Tr}
\DeclareMathOperator{\ad}{ad}
\DeclareMathOperator{\Lie}{Lie}
\newcommand{\g}{{\mathfrak g}}
\newcommand{\z}{{\mathfrak z}}
\newcommand{\s}{{\mathfrak s}}
\newcommand{\O}...

- 12.2k

2
votes

### Is it possible to average a riemannian metric over an action and preserve curvature bounds?

As it was mentioned by Igor Belegradek, the curvature gets destroyed by averaging.
Assume there is a modified the averaging process that preserves positive curvature.
Then you would prove Hopf ...

- 39.3k

1
vote

### Why is $\mathrm{SO}(4)$ not a simple Lie group?

The question is somewhat open-ended — more about beauty than truth. So I will offer two rather different narratives.
One way a group $G$ can fail to be simple is that it has a representation which is ...

- 50.1k

1
vote

### Counting adjoints in the symmetric or antisymmetric square of a Lie group representation

I don't have a proof but I noticed some things while having a quick look for some counterexamples that I thought were worth sharing. First things first what you are looking for is not special to the ...

- 747

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