# 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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### Flag manifolds as homogeneous Kahler manifolds

In this question it is asked if every flag manifold can be given the structure of a Kähler manifold. In the first answer it is written Flag manifolds exhaust all compact homogeneous Kähler ...
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### Fake degrees: why coinvariant algebra and classical groups over finite fields?

Apologies if this is not research level math (in that it concerns well-known stuff), but I am having trouble tracking down sources that explain the following. References would be very appreciated. ...
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### A geometric property of certain Lie groups

I call Poincaré $n$-half-space group the semidirect product of $\mathbb{R}^{n-1}$ and $\mathbb{R}^+$, where the action is by homotheties; equivalently as the group of translations and positive ...
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### Siegel Levi in $\operatorname{GSpin}(2n+1)$ and image into $\operatorname{SO}(2n+1)$

Let $T$ be a maximal torus of split $\operatorname{SO}_{2n+1}$ with basis $e_1, ... , e_n$. Let $$\Delta = \{e_1 - e_2, ... , e_{n-1} - e_n, e_n\}$$ be a set of simple roots of $T$ corresponding to a ...
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### Structure of extensions arising in Lie approximation of connected groups

My imperfect understanding is that, by the work of various authors (Gleason, Yamabe, Montgomery, Zippin ...), the following result is known: Let $G$ be a connected, locally compact, Hausdorff group, ...
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### Finding the closest special orthogonal matrix in Frobenius norm sense

Given a $3\times3$ matrix $M$, if we would like to get the closest $\mathrm{SO}(3)$ matrix $R$ that minimizes \begin{equation} \|R-M\|_F \end{equation} then $R$ = $UV^{T}$ where $U$ and $V^{T}$ are ...
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### De Rham cohomology of homogeneous spaces

Take a homogeneous space $K/L$, where $K$ and $L$ are compact lie groups. Denoting by $\Omega^*(K/L)$ its de Rham complex, which is a homogeneous vector bundle over $K/L$, and hence has a ...
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### Branching rules for E6 into SU(3)^3

I am very confused about what are the branching rules for representations of $E6$ into a $SU(3)\times SU(3)\times SU(3)$ subgroup. At least in the physics literature, there seems to be a serious ...
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### About the Cartan's moving frame method

I am learning Cartan's moving frame from chapter 6 of S.S. Chern's book "Lectures on differential geometry" (Google books). Suppose the moving frame in $E^N$ is denoted by $(p;e_1,\cdots,e_N)$, then ...
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### For $G$ an adjoint Chevalley group, are all of $G(\mathbb Z)$'s finite-index subgroups congruence subgroups?

Let $G$ be an adjoint Chevalley group. Are all of $G(\mathbb Z)$'s finite-index subgroups congruence subgroups? I read a theorem that states: When $G$ is the universal Chevalley group and it's not of ...
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### Identification problem: Does this group have a name?

I've encounter a group with properties that are very familiar, but I cannot say what group is it. Consider the variables $(t,x,y,z)$, an affine transformation $M \in A(3)$ on the last three variables ...
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### Is the Bruhat cell Zariski open in a connected algebraic group $G$? [closed]

Is the Bruhat cell Zariski-open in a connected algebraic group $G$? Specifically, is the big Bruhat cell Zariski-open (and maybe Zariski-dense)? Is it true for all the Bruhat cells?
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### Compact group actions with uniformly bounded derivatives

Suppose we have a smooth action of a compact Lie group $G$ on a non-compact smooth manifold $M$, denoted by $$\phi:G\times M\rightarrow M.$$ Differentiating $\phi$ at a point $x\in M$ gives a map that ...
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### How to prove that Chevalley groups over $\mathbb R$ have no compact factors

I am trying to see why the Chevalley groups (not limited to the adjoint group) over $\mathbb R$ are without compact factors in order to use the Borel density theorem. I've been told in another thread ...
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### An analytic vector field on $S^3$ whose all orbits are dense (à la Seifert conjecture, 2)

Does there exist a real analytic vector field on $S^3$ all of whose orbits are dense? The second paragraph of page 285 Of this paper says that there is a vector field whose almost all orbits are dense:...
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### Describing compact Lie groups in purely topological terms

Compact Lie groups are a very special type of compact group, namely those which admit a differentiable structure. Is it possible to describe compact Lie groups in purely topological terms, that is, ...
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### Derivative of expression involving a left-invariant connection

I'm trying to understand a calculation done in this paper. Somewhat simplified, the setup is as follows. Let $G$ be a Lie group, and $\varrho$ its Lie algebra. Let $\nabla$ be a left-invariant ...
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### Lie algebra of an elliptic curve

This might be a silly question, and if it has been asked somewhere else, I would appreciate a link; however, I was unable to find it myself. In this paper by Lauter-Viray (arXiv link), in the proof ...
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### Re asking:On the proof by Chu-Kobayashi that transformation groups are Lie groups

I have similar questions asOn the proof by Chu-Kobayashi that transformation groups are Lie groups and even more, how can $Y\in\mathfrak{g}^{*}$ generate 1-parameter global transformation group of $M$ ...
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### Character of a semisimple connected Lie groups [closed]

I'm trying to see why the Chevalley groups over $\mathbb C$ have no nontrivial character? I know that a compact connected semisimple Lie group has no nontrivial character but is the compactness ...
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### Amenability of the group of outer automorphisms of a connected compact Lie group

Forgive my ignorance on the Lie theory. I have the following questions in my current work concerning a certain property of compact connected Lie groups. First, allow me to fix some notations. Let $G$ ...