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Looking over some notes on on Lie Groups as relating to common matrix groups (i.e. $SO(N), SE(n), SO(N,1)$...), it's clear how the tangent space arises and how all elements of $G$ can be generated via the exponential map of the tangent space generators.

However, it is still not clear to me what purpose the Lie Bracket serves, why the commutator $[X, Y] = XY - YX$ specifically is used, and what the purpose the Jacobi identity has in this context. Is there an intuitive explanation of what the Lie Bracket "is doing", and the importance of the Jacobi identity? I can easily show that the commutator fulfills it, but it's not clear what the importance of it is.

Nathaniel Bubis
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  • For $[X,Y]=XY-YX$ see here, and the Jacobi identity says that $ad(x)$, defined by $ad(x)(y)=[x,y]$, is a Lie algebra derivation. – Dietrich Burde Jun 16 '23 at 15:00
  • @DietrichBurde I understand the technicalities. I'm trying to understand the motivation for the definitions. – Nathaniel Bubis Jun 16 '23 at 15:13
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    The motivation is given in the duplicate. This needs to be the Lie bracket for the tangent space at identity of the Lie group, in order to have an attached Lie algebra (which determines many properties of the Lie group just by linear algebra). The Jacobi identity then must hold, and it comes from differentiating with the Leibniz rule. – Dietrich Burde Jun 16 '23 at 15:16
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    If you think of a Lie group as an "infinitesimal" model of the group, then the Lie bracket is an infinitesimal version of the group commutator $[a,b]=aba^{-1}b^{-1}$ and the Jacobi identity is an infinitesimal form of the associativity axiom $a(bc)=(ab)c$. – Kajelad Jun 16 '23 at 16:08
  • Without the commutator (Lie bracket), every Lie algebra would just be a vector space, and every Lie group would be abelian. – Torsten Schoeneberg Jun 16 '23 at 23:53
  • @DietrichBurde - If you'd like to expand that as an answer that would be helpful, I'm not sure I understood the comment – Nathaniel Bubis Jun 18 '23 at 05:55
  • There are many good answers here already about this topic, and I do not want to repeat this. Have a look here at some posts. For a concrete example of the Lie group - Lie algebra correspondence, see here, or here, or here, and so on. – Dietrich Burde Jun 18 '23 at 08:54

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