# Tagged Questions

A Lie group is a group (in the sense of abstract algebra) that is also a differentiable manifold, such that the group operations (addition and inversion) are smooth, and so we can study them with differential calculus. They are a special type of topological group. Consider using with the ...

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### Need someone explain to me how to invert exponential map from SO(3) to so(3)

I need someone to explain to me how to invert exponential map from $SO(3)$ to $\mathfrak{so}(3)$. \begin{aligned} & R \in SO(3) \\ \text{From:}\\ \exp(\mathbf{M}) & = ...
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### Proof of Howe-Moore Property for SL(n,R)

On page 210 of Howe and Tan's Non-Abelian Harmonic Analysis there is the following proposition and proof: Let $(\rho, V)$ be a unitary representation of $SL(n,\mathbb{R})$. The following are ...
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### Identifying $\bar{D^3}$ with $S^3$ and then $\mathbb{R}P^3$ and $SO(3)$

After working it out, I found that we can identify the closed 2-disk, $\bar{D^2}$ with the 2-sphere, $S^2$. Let $\bar{D^2}$ have radius $\pi$. Then $\partial \bar{D^2}$ is identified with the south ...
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### Product rule of exponential matrix differentiation

Let $X,Y$ be two $n\times n$ complex matrices. Consider the function $f(t)=e^{tX}e^{tY}$. Is it correct that $\frac{d}{dt}f(t)=Xe^{tX}e^{tY}+e^{tX}Ye^{tY}$? Otherwise, how to prove that $X +Y$ is ...
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### Coadjoint representation $Ad^*$ of the Heisenberg group

The Heisenberg group $H^3$ is the set $\mathbb C\times \mathbb R$ endowed with the group law $$(z,t)\cdot(w,s) =\left (z+w, \,t+s+\tfrac{1}{2}\Im m(z \bar{w})\right).$$ For $z=x+ i y \in \mathbb C$ ...
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### The $Ad^*$ $G$-Orbits in $\mathcal{g}*$ (for $G$ being the Heisenberg group)

I did not understand why, the $Ad^*$ $G$-Orbits in $\mathcal{g}*$ (for $G$ being the Heisenberg group and $\mathcal{g}*$ is the dual space of its Lie algebra $\mathcal g$) are given by i) The ...
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### Lie bracket of a semidirect product

I'm trying to solve problem 1.12 of chapter 1 from Duistermaat & Kolk' Lie groups. In the exercise you have a Lie group $G$ and a finite-dimensional vector space $V$, and a homomorphism ...