# 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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### Conjugation to an element in a Bruhat cell with a Coxeter element

I would like to generalize a result in $\mathrm{SL}_n(\mathbb Z)$ to a general Chevalley Group $\mathrm{G}$.
In this article Meiri - Generating pairs for finite index subgroups of $\mathrm{SL}(n,\...

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### Reflection reverses a root strings

I am trying to understand the part of the proof In Fulton's and Harris's Represantation Theory book where he shows that the length of the root is at most 4:
Theorem If $\alpha,\beta$ are roots with $\...

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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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### Nontrivial relations of the irreducible root systems

For the root system of the type $A_n$, the roots are $\alpha _{i,j}$, $1\le i\neq j\le n$, we have the nontrivial relations $(x_{i,j} (t), x_{j,k}(u)) = x_{i,k}(tu)$ if $i, j, k$ are distinct. ($x_{i,...

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### Lattices in Lie groups

In the literature, people seem to predominantly look at lattices in nilpotent or reductive groups.
Is there a result that gives a general description of a lattice in an arbitrary Lie group?
Something ...

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### Analog of the Lie Product formula for commutators

Let $X, Y$ be elements of a Lie algebra. Consider the group $G$ generated by (limits of) arbitrary products of the elements
$$ G = \langle{e^{tX},e^{sY}\rangle}$$
for all $t,s$. The Lie product ...

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51 views

### Fierz identity for symplectic group

It is known that for the fundamental matrix representation of SU(N), with normalization given by
$$
{\rm Tr}(T^iT^j)=\frac{1}{2}\delta^{ij}
$$
there is a Fierz identity:
$$
\sum_{i=1}^{N^2-1}T^i_{ab}T^...

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### Almost complex structures on real Lie groups

Let $G$ be an even-dimensional Lie group, and let $\mathfrak{g}$ be its Lie algebra. I want to explore when a complex structure on $\mathfrak{g}$ induces a complex structure on $G$ making it into a ...

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### On the diameter of left-invariant sub-Riemannian structures on a compact Lie group

Let $G$ be a compact connected Lie group with Lie algebra $\mathfrak g$ of dimension $m$.
We fix an inner product $\langle\cdot,\cdot\rangle$ on $\mathfrak g$.
We may assume (in case is necessary) ...

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### Haar measure of the zero set of a nonconstant analytic function on a connected Lie group

Let $G$ be a connected Lie group equipped with its unique real analytic structure, $f : G \to \mathbb{R}$ a nonconstant real analytic function on $G$. Is the closed set $Z_f = f^{-1}(0)$ always of ...

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### Integral of Schur functions over $SU(N)$ instead of $U(N)$

Schur functions are irreducible characters of the unitary group $U(N)$. This implies
$$ \int_{{U}(N)}s_\lambda(u)\overline{s_\mu(u)}du=\delta_{\lambda\mu},$$
where the overline means complex ...

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### The Lie subgroup corresponding to inner derivations

Let $\mathfrak{g}$ be a finite-dimensional real or complex Lie algebra. We know that $Aut(\mathfrak{g})$ is a closed real or complex Lie subgroup of $GL(\mathfrak{g})$. We also know that the Lie ...

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### Coinvariant representative of homogeneous space cohomology

Given a compact homogeneous space $M = K/L$, consider its de Rham complex $(\Omega^*,d)$. Will every cohomology class $[\omega] \in ker(d)/im(d)$ contain a representative $\nu$ which is invariant with ...

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### Characterize an element of $\operatorname{SL}_n(\mathbb Z)$

I'm trying to generalize a theorem on $\operatorname{SL}_n(\mathbb Z)$ to the Chevalley groups over $\mathbb Z$.
In the theorem, there is a heavy use in the element $e_{1,n}(1)$ where
$$e_{1,n}(m)=
...

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187 views

### Existence of universal arrow from manifolds to forgetful functor of Lie groups

Let $M$ be a manifold, and $U$ be the forgetful functor from the category of Lie groups to the category of manifolds. My question is whether there is a universal arrow $(G, i)$ from $M$ to $U$? More ...

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### Representation theoretic characterisation of symmetric spaces

Let $G$ be a simple compact Lie group and $H$ a closed subgroup.
Let $\mathfrak{h}\subset \mathfrak{g}$ denote the corresponding Lie algebras. Let $\mathfrak{m}$ be an orthogonal complement to $\...

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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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### What is the connection between $\mathrm{AdS}_2$ and the hyperbolic plane $\mathbb{H}^2$?

What is the connection between $\mathrm{AdS}_2$ and the hyperbolic plane $\mathbb{H}^2$?
Some sources seem to imply that they are the same, i.e. having at least the same symmetry group $\mathrm{SL}(2,...

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### A matrix derivative

Suppose that $G$ is a connected Lie group of unitary matrices and $U(t)\in G$ depends continuously differentiable on a real parameter $t$ and has no real eigenvalue -1. Then the principal value of ...

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### Is every connected semisimple linear Lie group the connected component of (the real points of) an algebraic group?

Is every connected semisimple linear Lie group the identity connected component of (the real points of) an algebraic group?
I was told some fact along this line is true but could not find any ...

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### $L_\infty$-quasi inverse for the contravariant Cartan model on principal bundles

First of all I want to apologize for the much too long post.
A Lie group $G$ is acting on a smooth manifold $M$, then we define
\begin{align*}
T^k_G(M)=
(S^\bullet \mathfrak{g}\otimes T^k_\mathrm{...

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### Differential forms of a Lie group giving cohomology of the Lie group

Consider a manifold $M$. Then, we have the notion of differential forms on $M$ and complex associated to that, denoted by $$\cdots\rightarrow \Omega^{k-1}(M)\rightarrow \Omega^k(M)\rightarrow \Omega^{...

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### Infinitesimal description of homogeneous supermanifolds

Lie's third theorem says that if $\mathfrak{g}$ is a real, finite-dimensional Lie algebra, then there is a unique (up to isomorphism) simply-connected Lie group $G$ whose tangent Lie algebra is ...

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### Characterizing the big Bruhat cell of the universal Chevalley groups over $\mathbb C$

Is there a simple characterization of the big Bruhat cell of the universal (simply-connected) Chevalley groups over $\mathbb C$?
For example, it is known that the Borel subgroup of $\mathrm{SL}_n(\...

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### On maximal closed connected subgroups of a compact connected semisimple Lie group?

Let $G$ be a compact connected semisimple Lie group and let $\mathfrak g$ denote its Lie algebra.
Is the following result true? Does it follows directly from Dynkin's classification of maximal Lie ...

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### Moment map interpretation of Einstein equation

Einstein's famous equation relates the geometry of a (4-dimensional) manifold to the matter content in that manifold.
Is there a way to obtain Einstein's equation as a moment map?
More precisely, ...

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### Is there a weak homotopy equivalence between Sp(2n,ℂ)/U(n) and SU(n)?

This question, Is there a weak homotopy equivalence between Sp(2n,ℂ)/U(n) and SU(n)?, is at the end of a long string of my comments in
https://math.stackexchange.com/questions/3296373/is-sp2n-mathbbc-...

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### Subgroups of compact Lie groups generated by a subset of nodes of the Dynkin diagram

Where can I find a reference for the following fact, or as close as possible to it?
Let $G$ be a semisimple compact real Lie group with rank $r$, let $T$ be a maximal torus in $G$, let $\mathfrak{g}...

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### Idempotent functions on Sp(1)

The quaternion group $Sp(1)\simeq S^3$ can be understood as $(z,w)\in\mathbb {C}^2$ with $|z|^2+|w|^2=1$ where multiplication is defined by $(z,w)(t,s)=(zt-\bar{s}w,zs+\bar{t}w)$.
Question: How do ...

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201 views

### An application of Itô's formula to an SDE on a Lie group

I'm trying to understand a calculation in this paper (equation (3.8)). With some details removed, the setup is as follows.
Let $G$ be a Lie group, and $g(t)$ a curve in $G$ satisfying the SDE
$$dg(t)...

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### Descending central extensions to homogeneous spaces

Let $G$ be a Lie group (finite dimensional or Banach), and let $H$ be a Lie subgroup (in the Banach case we assume that $H$ is a submanifold which is also a Lie group). Let $\text{U}(1) \rightarrow \...

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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 $G$ ...