Talk:Adjoint representation

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I really wouldn't say [Cayley's theorem] has much to do with it. There is no obvious relation, for a group G, between acting on itself by conjugation and by translation - very different permutation representations.

Charles Matthews 06:43, 23 Aug 2003 (UTC)

Any Lie group is a representation of itself?

This does not sound right. "representation" can refer either to a vector space with an action of a group or to the group morphism from G to GL(V). Neither of these applies here.

Better: any Lie group G acts on itself...

Transport of tangent vectors?

I'm coming at this from a robotics/computer-vision perspective. I don't fully follow the Formal Definition section, but it sounds to me like the adjoint representation of a group element, g, Ad(g), is a linear operator that acts on the vector representation of a tangent vector at the origin, s the way g would operate on the matrix representation, S, of the same tangent vector at the origin. That is,

$S'=gSg^{-1}$

is equivalent to

$\mathbf {s} '=\operatorname {Ad} (g)\mathbf {s}$

up to representation. My sense is that $gSg^{-1}$ is S parallel transported from the tangent space of the identity to the tangent space of g, and so Ad(g) is doing a similar thing.

Is that even remotely right? Appologies for imprecise lingo. Thanks. —Ben FrantzDale (talk) 15:24, 18 May 2011 (UTC)

Derivative at the origin?

I'm confused by the sentence "It follows that the derivative of Ψ_g at the identity is an automorphism of the Lie algebra ${\mathfrak {g}}$." Which of these is it referring to (abusing notation)?:

1. $\left.{\frac {\partial \Psi _{g}}{\partial g}}\right|_{g=e}$
2. $\left.{\frac {\partial \Psi _{g}(h)}{\partial g}}\right|_{g=e}$
3. $\left.{\frac {\partial \Psi _{g}(h)}{\partial h}}\right|_{h=e}$

It's probably obvious if you understand this more than I do. :-) 15:56, 9 June 2011 (UTC)

Requested Move

The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section.

The result of the move request was: moved per request. Favonian (talk) 16:10, 21 February 2013 (UTC)

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Adjoint representation of a Lie group → Adjoint representation – Per WP:CONCEPTDAB, I believe this article should be located at Adjoint representation, which is currently a disambiguation page. The two concepts are so closely related, they are both defined in the article currently (the adjoint representation of a "Lie group" and a "Lie algebra"), so there is no need for a disambiguation page. Some textbooks define both concepts nearly simultaneously. This has also been discussed at WikiProject Mathematics. Mark M (talk) 17:14, 13 February 2013 (UTC)

• Support. The Lie group and Lie algebra points of view seem so closely connected that it makes sense to have a single article that covers both, and the new name better reflects this unity. —David Eppstein (talk) 18:29, 14 February 2013 (UTC)
• Support per nom and per David Eppstein, whose wisdom on this matter I trust implicitly. bd2412 T 16:31, 15 February 2013 (UTC)
  • Thanks, but please note that I am very much not an expert on Lie groups and Lie algebras. —David Eppstein (talk) 01:03, 17 February 2013 (UTC)
• Support -- Taku (talk) 01:35, 21 February 2013 (UTC)

The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.

Is really Ad : G -> Aut(g) for any Lie group G?

If Aut(g) are the automorphism of the Lie algebra g, that is linear operators preserving the Lie brackets, does Ad really associates to any element of G an element of Aut(g)? Is it possible that this holds only if G is a group of matrices for which holds the nice formula Ad_x (a) = x a x^(-1), (x is in G, a is in g), whilst in the most general situation G is sent by Ad into linear operators g -> g (not Lie bracket preserving)?

I'm probably wrong, it has been a while since i've studied this stuff. — Preceding unsigned comment added by 93.50.121.192 (talk) 22:32, 11 December 2013 (UTC)

Merger proposal

I note that Murata-san put up merge templates, but without starting, this, the merge proposal discussion. (WP:Merging). I'd let him do that here.If people did the merge carefully and tastefully, it would look like a good idea to me. (But I might not have the time to contribute to it, myself.) Cuzkatzimhut (talk) 15:25, 16 December 2014 (UTC)

Certainly. Actually, there was already discussion about merging adjoint representation of a Lie algebra to here, in the Lie-alg article. As you can see, the materials there mostly overlaps stuff here and there is not much to say about adjoint rep of Lie algebra that appear outside the context of Lie groups.

As for adjoint bundle, I don't see a need for a separate article as opposed to having a section.

-- Taku (talk) 22:28, 16 December 2014 (UTC)

Yes, if it were done with care, it is the most sensible thing to do. I'm just indicating that, procedurally, since all merge templates refer one to here, it is here, in the Target article, where the formal proposal and discussion should reside. Cuzkatzimhut (talk) 23:32, 16 December 2014 (UTC)

• Merge. Yes clearly. The only reason there are two articles is due to an old historical, uhh... accident. Creating the other article seemed like the right thing to do at the time. Oh, and I'd leave adjoint bundle alone. Right now, its thin, but I think it could be expanded ... bundles are really quite a different thing; half of particle physics happens on adjoint bundles, and you don't want to start talking about particle physics here, or, at least, not much. 67.198.37.16 (talk) 01:45, 18 September 2016 (UTC)
  • Given the consensus above that this is a good idea, I've moved the text over to about the right place, but haven't efficiently integrated it into the existing summary; some assistance in tidying it would be appreciated. Klbrain (talk) 09:12, 19 August 2017 (UTC)

Numerical Examples

Currently all the examples of adjoint representations in Lie groups are abstract. Can somebody provide a few numerical examples please? This would make the concept a lot easier to understand. Thank you! 23 January 2022 — Preceding unsigned comment added by 2603:7000:413F:9456:1128:6028:14AD:F69 (talk) 04:56, 24 January 2022 (UTC)

Another incomprehensible math article

In an introduction section where the words "for example" appear, it would be nice to see an actual example. Like here's a 2x2 matrix and it's adjoint. Look at the example that's given here in the 2nd sentence of the article and note how ridiculous it is. 98.156.185.48 (talk) 10:15, 8 July 2023 (UTC)