Talk:Lie algebra extension

Gilmore's book might have stuff (copied here from User talk:YohanN7)

... of relevance to incubated article. The last ref in group contraction by Gilmore has a neat section on Group expansions , III EXPANSIONS, pp 477—492, if you could get your hands on it. He believes in lots of hands-on examples and exercises.... Cuzkatzimhut (talk) 15:18, 14 May 2015 (UTC)

What should link here?

Surely these

• Group extension
• Virasoro algebra
• Witt algebra
• Semidirect product

Maybe these

• Heisenberg algebra
• Conformal field theory

Physics stack exchange link

Maybe some of universal covers and central extensions in QM can be worked into the article. YohanN7 (talk) 16:29, 25 April 2016 (UTC)

Extension by derivation

There are several things that I find curious on the subsection "6.3 By derivation". First I think e should be defined as h⊕g and not as a tensor product. Secondly, the bracket is weird : there is no mention of H1 and H2 anymore and as defined it is not bilinear : I think it shoud be [ aδ + G1, bδ + G2 ] = [G1,G2] + a δ(G2) - b δ(G1) Also, I dont think s as defined is a surjective morphism of algebra. Actually, e seems to be an extension of h by g and not of g by h. Finally, I don't see the difference between that and the extension by semidirect sum just above (just with h one-dimensional). Am I missing something ? --SylvL (talk) 16:19, 13 July 2017 (UTC)

Yes, yes, yes and yes. I fixed it. But as far as I can see, s, is at least surjective, and it should be easy enough to confirm (something I haven't done) that it is a Lie algebra morphism. I'll get back on the other points. YohanN7 (talk) 07:50, 14 July 2017 (UTC)

And yes again. The inclusion and surjection were upside down too. Does it make sense now? (Maybe a swap of notation, I. e. g ↔ h would be beneficial.) Thank you for pointing this out. YohanN7 (talk) 08:30, 14 July 2017 (UTC)

I think there is still an imprecision in the definition of the Lie bracket. It should be λδ and μδ in the bracket, no ? It might also be interesting to point ou that this is a particular case of the extension by semi-direct product.--185.30.25.218 (talk) 17:22, 14 July 2017 (UTC)

And yet again, yes (and oops). Given my track record in that section (and yours), it might be advisable that you edit the article yourself in the future? YohanN7 (talk) 08:50, 18 July 2017 (UTC)

Notation for null vector

I think the present notation for the null vector in section 4 (“Properties”) isn’t chosen well. It ought to be “0” (at the moment it looks like the notation for the empty set).

Projective representation

This section begins as following.

If M is a matrix Lie group, then elements G of its Lie algebra m can be given by

$G={\frac {d}{dt}}\left.(g(t))\right|_{t=0},$

where g is a differentiable path in M that goes through the identity element at t = 0. Commutators of elements of the Lie algebra can be computed using two paths, g₁, g₂ and the group commutator,

$[G_{1},G_{2}]={\frac {d}{dt}}\left.g_{1}(t)g_{2}(t)g_{1}(t)^{-1}g_{2}(t)^{-1}\right|_{t=0},\quad G_{1}=g_{1}'(0),G_{2}=g_{2}'(0).$

This seems to be wrong to me.

As far as I see,

${\frac {d}{dt}}\left.g_{1}(t)g_{2}(t)g_{1}(t)^{-1}g_{2}(t)^{-1}\right|_{t=0}=$

$G_{1}+{\frac {d}{dt}}\left.g_{2}(t)g_{1}(t)^{-1}g_{2}(t)^{-1}\right|_{t=0}=$

$G_{1}+G_{2}+{\frac {d}{dt}}\left.g_{1}(t)^{-1}g_{2}(t)^{-1}\right|_{t=0}=$

$G_{1}+G_{2}-G_{1}+{\frac {d}{dt}}\left.g_{2}(t)^{-1}\right|_{t=0}=$

$G_{1}+G_{2}-G_{1}-G_{2}=0$

since $g_{1}(0)=g_{1}(0)^{-1}=g_{2}(0)=g_{2}(0)^{-1}=1$

Am I right?

.

. 2A02:AB88:2A82:CD80:2099:559:7BEA:1261 (talk) 08:31, 17 August 2023 (UTC)

User 'n Quijote has modified this paragraph and has added a reference. I agree that it makes more sense to have two parameters s and t for the two different group elements. Still, the corrections have made the proof less clear. In addition, the commutator in Lie groups is NOT the adjoined but really the product G1 G2 G1^-1 G2^-1 (or in some notations G1^-1 G2^-1 G1 G2). Therefore, 'n Quijote, I am missing the last factor. Stefan Groote (talk) 07:26, 18 August 2023 (UTC)

Hi, thanks for this comment. Unfortunately, I didn't find any reference for the G1 G2 G1^-1 G2^-1 (could you show one please?) that's why I wrote G1 G2 G1^-1 (in accordance to Hall). Interestingly, this gives the same commutator.

Unfortunately, my modification really didn't make the things completely clear. The second equation, that is

$[U_{1},U_{2}]=\left.{\frac {d}{dt}}\right|_{t=0}\left.{\frac {d}{ds}}\right|_{s=0}U(e^{tG_{1}})U(e^{sG_{2}})U(e^{-tG_{1}})$

doesn't seem to be true for a projective-unitary representation, since in this case, $U(e^{-tG_{1}})=(U(e^{tG_{1}}))^{-1}$ my not be true. Can you fix this problem?

. 'n Quijote (talk) 05:31, 19 August 2023 (UTC)

Thank you for your addition. I can try to fix this problem. The commutator mathematicians are talking about in Lie group theory is shown in the very beginning of https://en.wikipedia.org/wiki/Commutator, and we of course should cite this in the article. Even though I do not have Refs. [1,2] at hand, I think that the books of Farleigh and Herstein cited on this wikipedia site contain the correct definition(s). Stefan Groote (talk) 07:04, 19 August 2023 (UTC)

Concerning $U(e^{-tG_{1}})=(U(e^{tG_{1}}))^{-1}$ I think that this should be clear because of the property that $U$ is a homomorphism (you just used it) and $1=U(e^{tG_{1}}e^{-tG_{1}})=U(e^{tG_{1}})U(e^{-tG_{1}})$. Stefan Groote (talk) 07:14, 19 August 2023 (UTC)

No, I'm talking about the non-homomorphism projective-unitary representation case. 'n Quijote (talk) 07:19, 19 August 2023 (UTC)

From https://en.wikipedia.org/wiki/Group_representation I read "a "representation" means a homomorphism from the group to the automorphism group of an object". Why do you need a non-homomorphic representation? Stefan Groote (talk) 07:23, 19 August 2023 (UTC)

This is the point of this section. Projective representations do not have always a unitary lift, only up to a cocycle. See Bargmann's "Unitary ray representations of continuous groups". 'n Quijote (talk) 07:27, 19 August 2023 (UTC)

I might be mistaken but unitarity is not necessary for what I have shown before. Anyhow, according to https://en.wikipedia.org/wiki/Projective_representation a projective representation is a homomorphism. Stefan Groote (talk) 07:35, 19 August 2023 (UTC)

The terminology is in this section somewhat misleading. Projective representations are really homomorphism to a factor group of an automorphism group. But we are here talking about projective-unitary (or in other words unitary ray) representations which are "representations up to a phase factor" (as swown in the text of this section) on the original (not factored) group (here on the unitary group). 'n Quijote (talk) 08:05, 19 August 2023 (UTC)

I've fixed this bug too. 'n Quijote (talk) 21:02, 19 August 2023 (UTC)