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I highly recommend this paper for a more in-depth treatment of exp/log map and the related representation of transformations: A Micro Lie Theory by Joan Sola,

I highly recommend this paper for a more in-depth treatment of exp/log map and the related representation of transformations: A Micro Lie Theory by Joan Sola, Jeremie Deray, Dinesh Atchuthan https://arxiv.org/pdf/1812.01537.pdf https://arxiv.org/pdf/1812.01537.pdf

JadeNB 4y ago
A warning, though: trying to switch from a mathematician's idea of Lie groups and algebras to a physicist's, or anyone else's, is not easy. I am a mathematician with a research specialty in Lie groups, and I remember being asked a question about this paper—I can't remember what it was, or I would be more specific—a while ago that was mathematically easy to answer, but where most of the time it took me to answer was spent in decoding the notation. This treatment is closer to the mathematician's than to the physicist's, but it still has some of the physics flavor about it—for example, it at least implicitly discusses Lie groups as if they come with a preferred action, which they need not—and, if you're not in this domain already, it's good to know what difficulties you can face later if you try to move between domains.
- andi999 4y ago
  From reading on mobile (so might have missed the clarification) it also seems to do the standard mistake in physics literature to assume the exponential map is surjective so the log map is defined on the whole group (which they call M?). This is not always true, I think sl2(R) is an example of a connected, non compact group such that the exponential map is not surjective.
  - chombier 4y ago
    Thanks, I just realized I was also assuming exp to be surjective as well, I stand corrected. The Wikipedia page[1] mentions the issue briefly but I was curious of the counter-examples. IIUC, matrices in SL(2) with trace < -2 have two distinct eigenvalues, one of which is negative[2], and such matrices cannot be reached by exponentiating elements of the Lie algebra sl(2) (traceless matrices). [1] https://en.wikipedia.org/wiki/Exponential_map_(Lie_theory)#Surjectivity_of_the_exponential https://en.wikipedia.org/wiki/Exponential_map_(Lie_theory)#S... [2] https://en.wikipedia.org/wiki/SL2(R)#Classification_of_elements https://en.wikipedia.org/wiki/SL2(R)#Classification_of_eleme...
    - chombier 4y ago
      I found this nice accessible proof for the interested: https://planetmath.org/slnrisconnected https://planetmath.org/slnrisconnected
      - andi999 4y ago
        Is this part of the proposition: x=exp X, and x having a double eigenvalue implying that X has a double eigenvalue somehow clear? How do you prove it?
- dimatura 4y ago
  Yeah, about 10 years ago I was reading one of the earliest papers applying these ideas to computer vision and I was pretty confused, as I never had any exposition to Lie group/algebras in my engineering undergrad. So I tried to read mathematical texts on this subject and I was 100x more confused! Would have been useful to article like the GP's one back then.

JadeNB 4y ago

A warning, though: trying to switch from a mathematician's idea of Lie groups and algebras to a physicist's, or anyone else's, is not easy. I am a mathematician with a research specialty in Lie groups, and I remember being asked a question about this paper—I can't remember what it was, or I would be more specific—a while ago that was mathematically easy to answer, but where most of the time it took me to answer was spent in decoding the notation. This treatment is closer to the mathematician's than to the physicist's, but it still has some of the physics flavor about it—for example, it at least implicitly discusses Lie groups as if they come with a preferred action, which they need not—and, if you're not in this domain already, it's good to know what difficulties you can face later if you try to move between domains.

andi999 4y ago
From reading on mobile (so might have missed the clarification) it also seems to do the standard mistake in physics literature to assume the exponential map is surjective so the log map is defined on the whole group (which they call M?). This is not always true, I think sl2(R) is an example of a connected, non compact group such that the exponential map is not surjective.
- chombier 4y ago
  Thanks, I just realized I was also assuming exp to be surjective as well, I stand corrected. The Wikipedia page[1] mentions the issue briefly but I was curious of the counter-examples. IIUC, matrices in SL(2) with trace < -2 have two distinct eigenvalues, one of which is negative[2], and such matrices cannot be reached by exponentiating elements of the Lie algebra sl(2) (traceless matrices). [1] https://en.wikipedia.org/wiki/Exponential_map_(Lie_theory)#Surjectivity_of_the_exponential https://en.wikipedia.org/wiki/Exponential_map_(Lie_theory)#S... [2] https://en.wikipedia.org/wiki/SL2(R)#Classification_of_elements https://en.wikipedia.org/wiki/SL2(R)#Classification_of_eleme...
  - chombier 4y ago
    I found this nice accessible proof for the interested: https://planetmath.org/slnrisconnected https://planetmath.org/slnrisconnected
    - andi999 4y ago
      Is this part of the proposition: x=exp X, and x having a double eigenvalue implying that X has a double eigenvalue somehow clear? How do you prove it?
dimatura 4y ago
Yeah, about 10 years ago I was reading one of the earliest papers applying these ideas to computer vision and I was pretty confused, as I never had any exposition to Lie group/algebras in my engineering undergrad. So I tried to read mathematical texts on this subject and I was 100x more confused! Would have been useful to article like the GP's one back then.