Talk:Table of Lie groups

Table layout

I'm having trouble parsing the following:

connected, not compact, for n≥2: not simply connected

I think that means that:

• For all n, its connected and not compact
• For n=1 it is simply connected
• For n≥2 it is not simply connected

Is that right?

I'm thinking of creating multiple lines for different values of n so that the if-then-else confusion can be eliminated. Am also thinking of replacing above with five columns:

• connected: y/n
• simply connected: y/n
• compact: y/n
• simple: y/n
• semisimple: y/n

This would make the table narrower and easier to read.

Is this a good idea?

I'm thinking of converting from html tables to wiki tables too.

I'm wondering if there should be a column that says

• nilpotent (refering to algebra)
• solvable (ditto)
• center

Everything listed so far seems to not be nilpotent. e.g. Heisenberg group is not listed.

Good or bad idea?

linas 04:30, 7 September 2005 (UTC)

I updated the table layout per above. Wow, was that ever tedious!

Listing S and SS as Y/N is a good idea, but I only see the 'Y'. I don't see the 'N'. Sure the reader can figure out that if left blank, it is equivalent to 'N'. but since you said Y/N, you really should do that instead of leaving it blank. It is not that tedious. And why are all the ones listed not nilpotent? How complete do you want this list to be? The impression I had on reading it was that it was meant to be pretty complete. If not, please say what the limits to completeness are intended to be.

For Real Lie Groups that have blank in the column for universal cover, does this mean they have none, or that the cover is uninteresting? From the Wikipedia page on universal covering groups, it would seem that they do have covers, since one would expect "locally simply connected" to imply "simple connected".

Finally, I don't think you really need centers: Cartan subalgebras would be more interesting. Or Dynkin diagrams, which you would probably want to represent by links, since they are covered elsewhere in Wikipedia. 64.105.137.254 (talk) 08:30, 16 July 2013 (UTC)

A template?

I wonder if it would be useful if we made a little template that can be placed at the bottom of each Lie group related page for navigational purposes? --HappyCamper 16:53, 18 August 2006 (UTC)

SE(3)

I am missing info on SE(3) = Lie group of rigid body transformations. It's a semi-direct product of SO(3) and R^3. --Benjamin.friedrich 13:42, 15 November 2006 (UTC)

Yes, I was looking for SE(3) too. Shouldn't that be included in the list of Lie groups? gnusbiz (talk) 20:10, 5 May 2008 (UTC)

Ditto. Why no SE(3)? —Ben FrantzDale (talk) 14:53, 17 May 2011 (UTC)

S(n) is not like the others

Do symmetric matrices belong in this table?

Every other group on this table comes from a Lie algebra of matrices with the Lie bracket given by commutator.

This one comes from take a vector subspace of matrices, decreeing that it has trivial Lie bracket, and then taking the Exponential map (Lie theory), and you can do this with any vector subspace of matrices; it's not an interesting construction, as far as I can tell.

The reason this concerns me is that symmetric matrices do not form a Lie algebra under the commutator; consider

$A={\begin{bmatrix}0&1\\1&0\end{bmatrix}}\qquad B={\begin{bmatrix}1&0\\0&0\end{bmatrix}}$

Then $AB-BA={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}$ is not symmetric.

Symmetric matrices naturally form a Jordan algebra, which is a nice object but complementary to Lie algebras.

Nbarth 00:23, 21 June 2007 (UTC)

It's indeed not like the others since the group multiplication given here is not the matrix multiplication. It is not an interesting construction in the following sense: You can take any manifold diffeomorphic to $R^{n}$ and give it a multiplication just by pulling forward (back) the multiplication (=addition) on $R^{n}$. What you get is only a Lie group isomorphic to $R^{n}$, so nothing new.

The symmetric positive definite matrices are a symmetric space of non-compact type diffeomorph to $GL(n,R)/O(n)$, but not a Lie group.

129.187.111.52 10:03, 21 September 2007 (UTC)

2-sphere?

Is the surface of a 2-sphere a Lie group? It seems like this would be a subgroup of SO(3), but I'm not sure... If it is, what's it called; if it isn't can someone explain why not? That is, isn't it the group generated by the matrix exponential like so:

$M=\exp {\begin{bmatrix}0&-z&y\\z&0&0\\-y&0&0\end{bmatrix}}$

Thanks. —Ben FrantzDale (talk) 01:40, 19 November 2009 (UTC)

No, this is not right. The only spheres that are Liegroups are the 1 and 3-dimesional (and the zerodimensional if you like). Unfortunately i can not explain exactly why this is the case. —Preceding unsigned comment added by 80.143.234.243 (talk) 13:16, 25 November 2009 (UTC)

OK. I'd love an explanation of why a 2-sphere isn't a group. I think that would help me understand exactly what constitutes a group versus a topological space. —Ben FrantzDale (talk) 17:12, 25 November 2009 (UTC)

A quick answer is that all lie groups have euler characteristic equal to zero. But an easy calculation shows that the euler characteristic of S² = 2. which excludes it from being a lie group. (The euler chracteristic is zero precisely for manifolds having a nowhere zero vector field. And its easy to create one of these on a lie group.) th2601:200:C082:2EA0:B9AD:9DB1:DF58:CB13 (talk) 21:36, 24 May 2023 (UTC)

I believe the Lie algebra so(4) is simple. Table is mistaken.

The table says the Lie algebra so(4) is an exception to the rule that the algebras so(n) are simple. I believe this is mistaken, perhaps arising from the fact that the group SO(4), considered as an abstract group, is not simple (it has a normal Z_2 subgroup). If someone has a definitively clear reference on this (I'll see if I can locate one) , let's fix it. MorphismOfDoom (talk) 15:52, 22 July 2013 (UTC)

I presume that by "algebras" you mean lie algebras.

SO(4) is isomorphic to S^3 x S^3 / 〈±1〉, whose lie algebra must be that of the double cover S^3 x S^3 (and hence a nontrivial direct sum). A lie algebra isomorphic to the direct sum of two lie algebras cannot be simple. (Here S^3 denotes the group of unit quaternions, or SU(2), or Spin(3), or Sp(1) — all of which are isomorphic.) — Preceding unsigned comment added by 2601:200:C082:2EA0:B9AD:9DB1:DF58:CB13 (talk) 21:33, 24 May 2023 (UTC)

Bad writing

The section describing the complex symplectic Lie algebra includes this phrase:

"complex matrices that satisfy JA + A^TJ = 0 where J is the standard skew-symmetric matrix".

Unfortunately the linked article about skew-symmetric matrices contains nothing about "the standard skew-symmetric matrix".

This is sloppy, bad writing. I hope someone familiar with the subject will define what J is in this Table of Lie groups article — but clearly this time.

What qualifies a Lie group to be included here?

I am thoroughly in favor of a "Table of Lie groups", even though it can't possibly include all of them.

It shouldn't be too hard to describe explicitly exactly what qualifies a Lie group or algebra to be included on this page.

If that it not the case, then I believe it would be much, nuch better if a simpler criterion were substituted for the current one for inclusion in this page. And also if a description of what qualifies a Lie algebra or Lie group for inclusion here were included in this article.

If, on the third hand, this article is just someone's recollection of his or her favorite Lie groups, then it makes good sense to delete the article from Wikipedia entirely, since personal points of view are expicitly excluded.