All Questions
13 questions
34
votes
4
answers
5k
views
Mathematical uses of string theory
It is widely believed that correctness of string theory as a physical theory will not be decided in the near future. Regardless whether this will turn out to be correct or not, mathematical concepts ...
29
votes
7
answers
8k
views
Topology of SU(3)
$U(1)$ is diffeomorphic to $S^1$ and $SU(2)$ is to $S^3$, but apparently it is not true that $SU(3)$ is diffeomorphic to $S^8$ (more bellow). Since $SU(3)$ appears in the standard model I would like ...
- 641
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
18
votes
2
answers
1k
views
Regarding Cayley Graphs of Property (T) Groups
A well-known application of Kazhdan's Property (T) is the construction of expander graphs. Background on this is discussed, for example, in this post on Terry Tao's blog. Essentially, Cayley graphs of ...
- 7,067
7
votes
3
answers
2k
views
Which Riemannian manifolds admit a finite dimensional transitive Lie group action?
This is a basically an adjusted version of my earlier question about how to define a convolution algebra on a general Riemannian manifold. The motivation for asking such a question of course comes ...
- 2,392
7
votes
2
answers
838
views
Transitive embedding of the projective plane $\Bbb R P^2$ into the $4$-sphere
Is there an embedding (i.e. injective continuous map)
$$\phi:\Bbb R \Bbb P^2\hookrightarrow S^4\subseteq\Bbb R^5$$
of the projective plane $\Bbb R\Bbb P^2$ into the $4$-sphere, that is ...
- 13.6k
7
votes
2
answers
3k
views
Why SU(3) is not equal to SO(5)?
I am asking in the sense of isometry groups of a manifold. SU(3) is the group of isometries of CP2, and SO(5) is the group of isometries of the 4-sphere. Now, it happens that both manifolds are ...
- 437
5
votes
1
answer
2k
views
Classification of discrete subgroups of the unitary group
Let $U(n)$ be the unitary group. From André Weil's paper "On discrete subgroups of Lie groups" it is well known that discrete cocompact subgroups of $U(n)$ have only a finite number of generators and ...
- 121
5
votes
0
answers
295
views
Unipotent representations of SL(2,R) by quantization
I'm a PhD student in mathematical physics and I happen to need some elements of Kirillov's "orbit method" for producing representations of Lie groups. I'm familiar with symplectic geometry, geometric ...
2
votes
0
answers
84
views
Weights of finite abelian group actions on submanifolds/subvarieties
(cross-posted from https://math.stackexchange.com/questions/4125529/weights-of-finite-abelian-group-actions-on-submanifolds-subvarieties)
How do weights associated to actions of finite subgroups of $\...
- 521
1
vote
1
answer
2k
views
Action of $SL(2,\mathbb{C})$ on representations of $SU(2)$
I want to precisely understand in what sense is (if it is!) $SL(2,\mathbb{C})$ the "complexified" version of $SU(2)$?
Can I think of it like choosing a natural matrix basis of the real three ...
- 3,541
1
vote
1
answer
204
views
Injective group homomorphism on $\frac{Spin(4k+2)\times U(1)}{\mathbf{Z}/2}$ or $\frac{Spin(4k+2)\times U(1)}{\mathbf{Z}/4}\to U(2^{2k})$
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SU{SU}\DeclareMathOperator\U{U}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}$From Pierre Deligne's Notes on spinors, we can see that there ...
- 317
1
vote
1
answer
608
views
Para-Complexification of Lie Groups
Let $G$ be a real Lie group. Then the complexification $G_\mathbb{C}$ of $G$ is the unique complex Lie group equipped with a map $φ:G\to G_\mathbb{C}$ such that any map $G\to H$ where $H$ is a
...
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