# Questions tagged [lie-groups]

Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.

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### Integral functional on algebraic varieties

Suppose that $X$ is a smooth complex algebraic variety and $X_{\mathbb{R}}$ is a real form of $X$. If $X_{\mathbb{R}}$ is compact and oriented as a real manifold, then it will admit a natural ...
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### Homotopically equivalent compact Lie groups are diffeomorphic

I have the following conjecture: Two homotopically equivalent compact Lie groups will be diffeomorphic. It may be necessary to restrict ourselves to only semisimple Lie groups. For simply connected ...
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### Viewing exceptional Lie algebras via the classical ones

I've been trying to understand the exceptional Lie algebras through the classical ones that I am more familiar with. In particular I wanted to get a handle on the root spaces and most discussions that ...
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### Erlangen program for “network geometry”

The subject of network geometry (Boguna et al., Network Geometry, Nature Reviews Physics 2021) looks at "geometric aspects" of complex networks. This is about studying a metric on the nodes, ...
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I've posted the following question some days ago in math.stackexchange https://math.stackexchange.com/questions/4155747/co-adjoint-orbit but I didn't get any answer! While I was trying to teach my ...
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### Cohomology group of a submanifold or Lie subgroup

In general: if one knows the cohomology group of some manifold ${\cal M}$, i.e. $H^n ({\cal M})$, are there known results for the same cohomology group $H^n (X)$ of a submanifold $X \subset {\cal M}$? ...
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### Three dimensional real Lie groups with cocompact discrete subgroups

I would like to know what are all the real three dimensional Lie groups (simply connected) that can act transitively and locally freely on a compact three dimensional manifold? This is equivalent to ...
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### Classification of the group action

Let $G$ be a closed subgroup of $O(n)$ such that $\mathbb R^n/G$ is isometric to $\mathbb R^{n-2} \times \mathbb R_+$. Can we have a classification of $G$ up to conjugation?
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### Geodesics on algebraic manifold

A nonsingular algebraic manifold is an immersed manifold (slightly different from the usual algebraic manifold which is required to be embedded) $M \subseteq \Bbb{R}^n$ (with induced topology) that is ...
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### $G/T$ has finitely many $G^\theta$ orbits

Let $G$ be a compact connected Lie group and T be it's maximal torus. Let $\theta: G \rightarrow G$ be an involution on $G$ and let $G^\theta = \lbrace g \in G , \theta(g)=g \rbrace$. I'm looking for ...
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### Extending group actions to vector bundles

Let $G$ be a group acting on a manifold $M$. Suppose $V$ is a rank $n$ vector bundle on $M$. Is there any obstruction to extending the action of $G$ to $V$? In how many ways can the action be extended ...
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### Properties of stabilizers of adjoint action general linear group

Let $G=GL(n,\mathbb{C})$ and let us consider $x \in GL(n,\mathbb{C})$. I'd like to know whether the following is true: the stabilizer for the conjugation action $C(x)$ is special in the sense that ...
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### A specific coset decomposition of $\mathrm{GL}_n(\mathbb{C})$

Disclaimer: I am a theoretical chemist (not a mathematician). I have tried asking this question at Math SE with no luck (https://math.stackexchange.com/questions/4080696/a-specific-coset-decomposition-...