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I m reading a paper involving Lie algebra, and there are some definitions that I don't understand.

$\mathcal{M}$ is the smooth manifold, and $\mathcal{X} \in \mathcal{M}$ is a point of a Lie group on the manifold. $\mathcal{X}^{-1} \mathcal{X} = \mathcal{E}$ defines the unit element and the inverse of point $\mathcal{X}$ in the group. $\mathcal{T}_\mathcal{X}\mathcal{M}$ is the tangent space at point $\mathcal{X}$, and $\mathbf{v}^\wedge$ is the element of this Lie algebra. The derivative of the unit element $\mathcal{E}$ gives a constraint $$\dot{\mathcal{X}^{-1}}\mathcal{X} + \mathcal{X}^{-1}\dot{\mathcal{X}} = 0 \qquad (Eq1)$$ Another form of Eq1 is simply $$\dot{\mathcal{X}^{-1}}\mathcal{X} = -\mathcal{X}^{-1}\dot{\mathcal{X}} \qquad (Eq2) $$ Then, the paper gives directly a definition that $$\mathbf{v}^\wedge = \dot{\mathcal{X}^{-1}}\mathcal{X} = -\mathcal{X}^{-1}\dot{\mathcal{X}}\qquad (Eq3)$$

What I am confused is that $\mathbf{v}^\wedge$ is the velocity and is an element of the Lie algebra or tangent space. Or, $\mathbf{v}^\wedge$ should be the velocity at a specific point if we give it by $^\mathcal{X}\mathbf{v}^\wedge$ or $^\mathcal{E}\mathbf{v}^\wedge$. At least, $\mathbf{v}^\wedge = \dot{\mathcal{X}}$ for me. So, why it is defined as Eq3 or how Eq3 is derived?

MakeItGreatAgain
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  • I have trouble understanding your question. Is $\mathcal{M}$ suppose to be a Lie group? What is $\textrm{v}^{\wedge}$? Is it defined by Equation 3? Also, what do you mean by "the derivative of the unit element $\mathcal{E}$"? Which function are you differentiating exactly? – Jacques Sep 19 '24 at 14:39
  • @Jacques Thanks for reply. Actually, there was a similar question after I posted this. There was no clear answer as well in that post. $\mathcal{M}$ is the manifold and $\mathcal{X}$ is said to be a point in the Lie group of $\mathcal{M}$. They differentiate $\mathcal{X}^{-1}\mathcal{X} = \mathcal{E}$. Two links almost tell what I don't understand: https://math.stackexchange.com/questions/4602758/how-to-get-the-form-of-elements-of-the-lie-algebra?rq=1 , https://math.stackexchange.com/questions/4546616/find-structure-of-lie-algebra-from-lie-group?rq=1 – MakeItGreatAgain Sep 20 '24 at 03:07
  • What do you mean by the "Lie group of $\mathcal{M}$"? – Jacques Sep 20 '24 at 10:29
  • @Jacques, $\mathcal{M}$ is a Lie group's manifold. – MakeItGreatAgain Sep 24 '24 at 07:34
  • Can anybody help a little bit for this? – MakeItGreatAgain Sep 25 '24 at 05:42
  • I can't understand your question. Some statement are unprecise, like "the derivative of the unit element" (you can take the derivative of a function, not of an element). Also why is $\textrm{v}^{\wedge}$ "$\textbf{the}$" element of this Lie algebra? Is it just $\textbf{an}$ element? But most importantly, it is not clear what you want to know. I think you should edit your question and rephrase it. – Jacques Sep 25 '24 at 08:20

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