# Tagged Questions

1answer
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### What makes a Lie Group a Differentiable Manifold?

I've recently been trying to glance at the definition of a Lie group, but I'm not clear as to why a Lie group is defined the way it is, and why this becomes a smooth manifold. For example, if we have ...
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### Affine connection on a Lie group.

Let $G$ be a Lie group. For $g \in G$, we can define a diffeomorphism $l_g: G \to G$ by $l_g(x)=gx$, and a bundle map ${l_g}_*:TG \to TG$. Then, I guess that we can obtain the affine connection on $G$ ...
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### Special conformal killing fields - solving for integral curves.

For each $b\in\mathbb R^d$, let a vector field $X_b:\mathbb R^d\to\mathbb R^d$ be defined as follows: \begin{align} X_b(x) = 2(b\cdot x)x - x^2 b, \end{align} where $x^2 = x\cdot x$. This is the ...
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### The diffential of commutator map in a Lie group

Leb $G$ be a Lie group and $f:G\times G\rightarrow G$ be the commutator map $:(x,y)\mapsto xyx^{-1}y^{-1}$. How to obtain the Lie bracket in the associated Lie algebra of $G$ from the derivatives of ...
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### Show that an Ehresmann connection on a principal G bundle is equivalent to a Lie Algebra Valued one form.

Let $E$ be a smooth principal $G$-bundle on M. The vertical bundle $V$ is defined as $V=\ker(d\pi:TE\to \pi^*TM)$. An Ehresmann connection on $E$ is a smooth subbundle $H$ of $TE$ (also called the ...
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### Definitions of Semisimple Lie Algebra

We usually define semisimplicity of a Lie algebra $\mathfrak{g}\subset M_n ({\bf R})$ from two ways. I want to know the relation between them. One of the definitions of semisimple Lie algebra is ...
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### Compactness of semisimple Lie algebra

I want to prove that on a semisimple Lie algebra $\mathfrak{g}$ over ${\bf R}$: $\mathfrak{g}$ is compact if and only if the Killing form is strictly negative definite. Here the Lie algebra is ...
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### Derivations on semisimple Lie algebra

First recall some definitions : Let $B$ be a Killing form on Lie algebra $\mathfrak{g}$ over ${\bf R}$ such that $B(X,Y)\doteq Tr(ad_Xad_Y)$. $\mathfrak{g}$ is semisimple if $B$ is ...
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### Tangent Space of $\operatorname{Aut}(T_e G)$

Let $G$ be a Lie Group, how to prove that the tangent Space of $\operatorname{Aut}(T_e G)$ at identity is $\operatorname{End}(T_e G)$? Thanks.
2answers
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### Two Definitions of the Special Orthogonal Lie Algebra

I am encountering two definitions of the special orthogonal lie algebra, and I would like to know if they are equivalent, and if there are advantages to working with one over the other. If we begin ...
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### Invariant Inner Product on Lie Algebra

Let $G$ be a Lie group, $\frak{g}$ its Lie algebra. Suppose $\mathcal{D}$ a representation of $G$ on $V$, $d$ the associated Lie algebra representation. Suppose $V$ is endowed with an inner product. ...
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### The set of complete vector fields

The set of all complete vector fields in $\mathbb R^{n}$ is closed under Lie bracket? is this set a $D$-module where $D$ is the ring of bounded smooth funcions? Can anyone recomend me a book on the ...
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### Induced metric via $\mathbb C P^n \cong SU(n + 1)/S(U(n) + U(1))$

I was wondering if the homeomorphism above gives me the Fubini Study metric on $\mathbb C P^n$. More precisely: Consider $\mathbb C P^n$ equipped with the metric induced by the standard construction ...
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203 views

### jacobian involving SO(3) exponential map: $\log(R * \exp(m))$

I would like to compute the 3x3 Jacobian of $$\log(R * \exp(m))$$ with respect to the 3-vector $m$, evaluated at $m=0$. In the above, $\exp$ is the exponential map from so(3) to SO(3), $\log$ is ...