# Questions tagged [lie-groups]

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

1,982
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### A geometric property of certain Lie groups

What is an example of a Lie group $G$ not ismorphic to the Poincare upper half space $H^n$ but satisfy the following:
For every left invariant metric $g$ we draw all geodesics. Then we CAN rescale $...

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33 views

### 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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53 views

### 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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59 views

### 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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49 views

### 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,...

**5**

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82 views

### 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 ...

**5**

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**1**answer

100 views

### 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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79 views

### 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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109 views

### 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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51 views

### 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 ...

**5**

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160 views

### 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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51 views

### 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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91 views

### 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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133 views

### 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)=
...