Talk:Lie algebra/Archive 1

Text from 2001

Why is it called a Lie algebra?

---

What are Lie algebras used for? How do they relate to more usual objects, such as groups?

---

Sophus Lie developed the theory for solving partial differential equations. Lie algebras are built on Lie groups.

Definition in char 2

Our definition of Lie algebra currently appears to be slightly wrong. If the base field has characteristic 2, then xy+yx=0 and x²=0 are not equivalent identities, and it's surely x²=0 that is required. I haven't attempted to fix the article, since there appears to be a new Wikipedia bug that corrupts HTML character entity references. --Zundark, 2002 Feb 7

It seems this bug only affects the preview, so I've modified the article now. --Zundark, 2002 Feb 7

Lie algebra whose bracket is identically zero(Example 1)

i think it is completely fair to call a Lie algebra whose bracket is identically zero "uninteresting". it is a trivial Lie algebra.

It was right to change "uninteresting" to nothing or to "trivial", because "uninteresting" on its own, thrown into an article on algebra, sounds like a schoolboyish comment on how boring it is.

Now who is going to change the german version of this page?

Just to be clear, i think you are right, the phrase "very uninteresting" doesn't sound very encyclopedic. "trivial" is better. on the other hand, it doesn't sound schoolboyish to me, i have heard methematicians use this very language to describe things like 0-dimensional vector spaces, various trivial things. so I am pretty sure it was not vandalism, which is why i initially reverted your change. whether or not it is the most appropriate choice of wording, well, that is a different matter. Lethe

I understand. I didn't mean to say it was necessarily vandalism, just that it looked like it.

• Is the Lie group for this trivial algebra the nonzero elments of the base field?Thanks,Rich 20:24, 31 October 2006 (UTC)
• Or maybe it would be just the one element group {1} of the field? (Since there are probably lots of less 'brutal' Lie brackets than the triv one here that still make for commutative Lie algebras and so give an abelian lie groups-I bet it's one of those less brutal brackets that gives rize to the nonzero scalars as the Lie group.)Rich 21:02, 31 October 2006 (UTC)

Clarification requested

Could someone clarify the following definition:

[X, Y] f = (XY − YX) f for every function f on the manifold  ?

I don't see what XY is. PJ.de.Bruin 18:26, 12 Jul 2004 (UTC)

A vector in differential geometry is a directional derivative evaluated at a point, which therefore acts on a smooth function on the manifold and returns a number X_pf. A vector field is a directional derivative of a function X(f) which is also a smooth function (just like the derivative of sin x is also a smooth function).
Therefore you can take a second directional derivative, with respect to a different direction Y, and get Y(X(f)). The second order differential operator Y X is not a vector field (vector fields must be first order differential operators by definition), but the combination XY−YX is, because the second order terms cancel out, due to equality of mixed partial derivatives.

In local components, if X=X^a∂_a and Y=Y^b∂_b (using the Einstein summation notation), then

$[X,Y]f=(XY-YX)f=X(Y(f))-Y(X(f))=X^{a}(\partial _{a}Y^{b})(\partial _{b}f)+X^{a}Y^{b}\partial _{a}\partial _{b}f-Y^{b}X^{a}\partial _{b}\partial _{a}f-Y^{b}(\partial _{b}X^{a})(\partial _{a}f)$ .

The middle two terms there will cancel, leaving only first order derivative terms of f.

So in short, the notation XYf means Y acts on the function, taking the directional derivative, and returning a smooth function, and then we let X act on the resulting smooth function. This is more clearly brought out when you use the notation X(Y(f)), however people often leave off the parentheses in this context, because they become cumbersome.

One could also take the f out of the equation, switch the dummy index in one of the terms, and get

$[X,Y]^{b}=X^{a}\partial _{a}Y^{b}-Y^{a}\partial _{a}X^{b}$ ;

this notation would be common in, for example, a GR textbook.

In my opinion, explanations of these terms should not necessarily be included in an article on Lie algebra. Lie algebra is an application of linear algebra, and discussions of differential topology are somewhat out of context. It is interesting that the space of vector fields from differential topology is also a Lie algebra, so it is good that that is mentioned as an important example of a Lie algebra in this article, but further explanations of these terms should probably be in differential topology or vector field (although I don't see this explanation in either of those two articles. It should be added somewhere).

-Lethe 21:21, Jul 12, 2004 (UTC)

Vector fields as operators

[X, Y] f = (XY − YX) f for every function f on the manifold

(here we view vector fields as operators that turn functions on a manifold into other functions).

Could someone remind the naive reader (me) how this is done? I have a vague recollection: X f = the directional derivative of f in the direction of X, so that X is in effect a partial differential operator; then XY means the composition of partial differential operators. Is that right? Michael Hardy 23:24, 24 Sep 2004 (UTC)

yes, XY is the composition of the two derivative operators, making it a second order derivative, and therefore itself not a vector. only the difference XY-YX is a first order derivative operator -Lethe | Talk

also, i noticed you added the word "smooth" there. I added another instance; we usually require the vector fields to be smooth as well. however, strictly speaking, we only need f to be C² and the vector fields to be C¹. I'm not sure it matters enough to change it though, and most textbooks usually assume smoothness for simplicity. -Lethe | Talk

Category-theoretic definition

The category theoretic definition provided doesn't work over a field of characteristic 2. The correct definition is as follows:

A Lie algebra is an object A in the category of vector spaces together with a morphism $[\cdot ,\cdot ]:A\otimes A\rightarrow A$ such that

$[\cdot ,\cdot ]\circ \Delta =0$ where $\Delta :A\rightarrow A\otimes A$ is the diagonal morphism, and

$[\cdot ,\cdot ]\circ ([\cdot ,\cdot ]\otimes id)\circ (id+\sigma +\sigma ^{2})=0$ where σ is the cyclic permutation braiding $(id\otimes \tau _{A,A})\circ (\tau _{A,A}\otimes id)$.

(It should probably also be added that the morphism $\tau _{A,A}:A\otimes A\rightarrow A\otimes A$ is the interchange morphism rather than leave that as assumed knowledge.)

I didn't want to edit the given definition because I don't want to get rid of the diagram, which is correct apart from the case of characteristic 2.

Could we put this category theoretic definition towards the end? To me this notation seems almost as bad as Bourbaki's notation for quantifiers. (Yeah I know it has a glorious tradition — Frege and all that) CSTAR 00:08, 30 Dec 2004 (UTC)

I think it also needs some indication of which id is which here. Kuratowski's Ghost 15:09, 7 Apr 2005 (UTC)

My impression is that whenever you write something like $f\otimes g:A\otimes B\rightarrow A\otimes B$, the implicit meaning is essentially that f acts on things in A and g acts on things in B. So I don't share your confusion with this aspect of the notation, although I do agree with the great-grandparent in this thread that the notation is rediculously unclear.

Killing vector field?

Good god, this section is incomprehensible! You are calling a left invariant vector field a Killing vector field? This is going to generate an enormous amount of confusion with Killing forms. Also the explanation is ridiculusly complicated. True the left action of G on itself allows to transport the tangent space at to any tanget space on the group. (and we don't have to say without holonomy). Somebody with time please rewrite this!

And what's this conversational style? Let's say ...blah.CSTAR 15:33, 18 Jan 2005 (UTC)

Is this a mixup between LIVFs and Killing vectors? LIVF are already mentioned early in the article, so...? -Lethe | Talk 15:42, Jan 18, 2005 (UTC)

Killing vectors of Lie algebras

User:Phys, can you help me understand this recent addition of yours?

"for each element of the tangent space of G at the identity e, there naturally corresponds a Killing vector field over G generated by the regular representation of G upon itself (Take a differentiable parametrized path passing through the identity and take the derivative at the identity)."

firstly, as far as I am aware, Killing vectors depend on the existence of a metric. I know that the Killing form of a semisimple Lie group gives rise to a Riemann metric, but the non-semi-simple case? I didn't follow how the regular representation gives you a metric...? -Lethe | Talk 15:37, Jan 18, 2005 (UTC)

The new text follows here for reference. I have removed it from the article, until its validity can be established. Lethe | Talk 03:24, Jan 21, 2005 (UTC)

Lie algebras from Lie groups

Let's say we have a Lie group G. for each element of the tangent space of G at the identity e, there naturally corresponds a Killing vector field over G generated by the regular representation of G upon itself (Take a differentiable parametrized path passing through the identity and take the derivative at the identity). From differential geometry, we have the Lie bracket (see Lie derivative) between any two vector fields. It turns out the Lie bracket of the two Killing vector fields generated by any two elements of the tangent space at the identity is another Killing vector field generated by another element of the tangent space at the identity. It turns out this has the structure of a Lie algebra.

Examples

The example given here is not t h e Heisenberg Lie algebra, since that is defined for arbitrary symplectic vector spaces (of dimension 2n). The three nilpotent matrices given here, are the special case of a 2-dimensional symplectic vector space only, represented by these three matrices. This representation can be generalized to Heisenberg Lie algebras in the higher dimensional case. — Preceding unsigned comment added by 130.133.134.16 (talk) 16:31, 27 September 2011 (UTC)

dual algebra

Is the vector space dual of a Lie algebra naturally a Lie algebra? And the tensor product of two Lie algebras? If so then Hom(g, g) would also be a Lie algebra if g is and the connection form could take values therein. --MarSch 11:48, 19 October 2005 (UTC)

No, the dual and tensor of Lie algebras are not natually Lie algebras. -Lethe | Talk 13:29, 19 October 2005 (UTC)

So what about the automorphisms of a Lie algebra? --MarSch 14:22, 19 October 2005 (UTC)

That's a group. If the Lie algebra is a finite-dimensional Lie algebra over R, then it's a closed subgroup G of GL and hence a Lie group. The Lie algebra of the group G is the Lie algebra of derivations of the original Lie algebra (Umm I'm giving a quick answer so the last statment might require some modification).--CSTAR 14:27, 19 October 2005 (UTC)

Okay. What I want to know is what Lie algebra a connection form for a G-bundle is valued in, since that article says it is Hom(F, F) for fiber F, which makes sense but perhaps the algebra product generates Aut(F)? Or perhaps a connection form is not a Lie algebra form?--MarSch 15:30, 19 October 2005 (UTC)

The connection form of a G-bundle is g-valued, where g is the Lie algebra of G. The part you mention is not for G-bundles though, it's for the associated vector bundle. A vector bundle can be associated with a GL(n)-bundle. The Lie algebra of GL(n) is gl(n). gl(n) is another way of writing Hom(F,F), where the fibre F is an n-dimensional vector space, not a Lie group. Incidently, I guess Hom(g,g) is a Lie algebra. Simply define [S,T](X) to be ST(X)-TS(X). I don't think that construction is too useful though , and it's still true that the dual and tensor are not Lie algebras, and it's not what's meant in the connection form article, where F is a vector space, not a Lie algebra. -Lethe | Talk 19:58, 19 October 2005 (UTC)

I suspect the question is based on a misunderstanding, and so has a flawed answer. Lets say we have a connnection on a fiber bundle P whose fiber is a Lie group G, over a base manifold M. Now pick a vector field X on M, and pick a point p in P. Then the connection A for vector field X at point p will just be an element of the lie algebra g of G. There is no dual. The "form" part of the "connection form" refers only to the base manifold, and not to the lie group. Does that answer the qeustion? linas 23:34, 19 October 2005 (UTC)

To answer his question, you should probably address how Hom(F,F) is supposed to be a Lie algebra. I think that's why he's bringing up duals and tensors. He's thinking of the isomorphism Hom(F,F) = F*xF. His real problem is that he is confusing the vector space F on which a rep of g acts with the Lie algebra g itself. It becomes clear once you distinguish between a principle G-bundle and an associated vector bundle. -Lethe | Talk 00:23, 20 October 2005 (UTC)

Here's a way to think about it: suppose M is two-dimensional, and suppose G is 9-dimensional. Then the connection form has one index that runs over 1,2 and two more indecies that run over 1..9. The two indecies running over 1..9 are a matrix, and that matrix is an element of the Lie algebra. The index that runs over 1,2 is the "form" part of the connection form, its the part that is to be contracted with a vector field X on M. (the tangent space to G is g, and one does not need to use forms to move around on G, it is enough to use the group action). linas 23:47, 19 October 2005 (UTC)

Of course, you only get a matrix once you choose a representation (and thereby move to the associated vector bundle). And then it need not share the dimensionality of the Lie group. For example, an associated SU(2) vector bundle using the fundamental rep will be a 2x2 matrix valued form, even though SU(2) is 3-dimensional. -Lethe | Talk 00:23, 20 October 2005 (UTC)

Respons to the 2-dim M, 9-dim G example. This means that g is also 9-dim, so why would the g-valued connection form have two indices running from 1 to 9? Perhaps we can choose an example to work out and add to connection form which does a very bad job of making things clear which is why I added the expert tag. --MarSch 10:21, 20 October 2005 (UTC)

Actually remembering that A is really an array of LAv-forms I can understand this, but the example would still be appreciated. --MarSch 10:32, 20 October 2005 (UTC)

And the representation of the Lie algebra is the isomorphism g ~=~ Aut(V) = Hom(V, V), which explains another way of turning one index into two. So why do christoffel symbols only have 3 indices. One form-index and 2 from Aut(TM) I guess, since they all run from 1 to 4 (0 to 3) or is the form-index suppressed? What happened to the array index? --MarSch 10:50, 20 October 2005 (UTC)

Aut(V) is the group of isomorphisms, Hom(V,V) is the algebra of homomorphisms. These things are never isomorphic. (For example, End(V) always contains 0, Aut(V) never does). Elements of Aut(V) and End(V) will both have 2 indices when written in coordinates, so changing from one to the other won't get rid of indices. The Christoffel symbols have 3 indices like all connection forms. 2 indices indices for g (so for Christoffel), and one for the one-form. -Lethe | Talk 20:19, 20 October 2005 (UTC)

As I found Lethe's remarks below confusing, my best recommendation is to cite a book that is quite readable: Jurgen Jost "Riemannian Geoemtry and Applicatons" (do not confuse with his similarly named book "Compact Riemann Surfaces"). I've been reading this recently, and (to me at least) its the clearest, best-constructed and most approachable book on the topic that I've seen. (And yes, it seems that gauge theory is now a part of reimannian geometry these days, so he explains Yang-Mills at the same time, without any big to-do.). linas 23:37, 20 October 2005 (UTC)

Are you talking about ISBN 3-540-42627-2, Riemannian Geometry and Geometric Analysis? I couldn't find another Riemannian geometry book by Jost on Amazon, so I'll assume that's what you're talking about. Anyway, looking through the table of contents of the book online, I see that he only treats vector bundles, makes no mention of principle bundles or more general bundles. I also don't see Yang-Mills in the table of contents, so maybe I've got the wrong book. Anyway, I'm disappointed that you didn't find my list elucidating. I thought by expressing everything in local coordinates, I was being as down-to-earth and concrete as one can get. -Lethe | Talk 00:15, 21 October 2005 (UTC)

Sorry, I didn't mean to dis your work; it looks good but my eyes glazed over. Yes, that is the book. The chapter titles are misleading, the book is a bit deeper than the titles would indicate; but its not a book about fiber bundles per-se. Mostly I just thought the treatment was refreshing in that it used the general language of bundles, introducing F and A first (even on vector bundles) and then showing how the riemann curvature R is related to F when the fiber is a vector bundle. When I first learned this stuff, the textbooks didn't do this, so it was very hard to figure out how these concepts were similar/the same, as superficially, they seemed to be so different (i.e. using F for lie groups, using R for geometry). linas 10:34, 21 October 2005 (UTC)

I know what you mean. I just figured out that they are related too. Unfortunately GR has a different Lagrangian than Yang-Mills. Do you happen to know what happens when you choose the same Yang-Mills Lagrangian for GR? Is it in accord with experiment? Is there an article yet? --MarSch 11:35, 21 October 2005 (UTC)

I think the YM langrangian and the EH lagrangian are as close to being the same as they can be, or at least as they should be. YM is tr(F ^ *F) , while EH is tr(R ^ *(θ^θ). The difference is in that second term, θ^θ (here θ is a bundle morphism from a Lorentz bundle to our Riemannian spacetime aka vielbein). We can include it here because it makes the action into a guage invariant scalar. We do not have that corresponding structure available in YM theory (which is one way that makes YM theory different from GR in a basic way). Including that term has an important effect: it makes the connection not an independent dynamical variable. If you used R^*R as your action, you'd have YM theory on your connection, which would mean that your connection is no longer the Levi-Civita connection, and your vector bosons would be spin-3. I believe its known that there cannot be interacting theories of spin 3 massless particles. The EH action is known as the only action in the curvature which satisfies some physical requirements like the equivalence principle and vanishing of torsion in the vacuum. But actions higher order in the curvature (which still do not have the connection as independent) have been considered in the literature. String theory, for example, predicts higher order terms in the curvature, so models based on string theory generally have more general actions. -Lethe | Talk 17:29, 21 October 2005 (UTC)

New article

Gaack. For better or worse, there is now a brand-new article called Gauge covariant derivative that attempts some sort of low-brow review of this topic. Its maybe an over-simplified presnetation but thought I'd mention it. linas 23:50, 20 October 2005 (UTC)

I think reading this just caused me to lose my mind. —Preceding unsigned comment added by 68.144.103.61 (talk) 03:56, 15 April 2008 (UTC)

connection forms in coordinates

I'll use A for the connection form components of a principle bundle in local coordinates, and Γ for the covariant derivative components of a vector bundle.

principle bundle

on a principle G-bundle, given a local trivialisation, a connection form in local coordinates can be written as a g-valued one form on the base space. Therefore, it will look like

$A=T^{r}\otimes A_{r\mu }dx^{\mu }$

where {T^r} is a basis for g, and {x^μ} are local coordinates for the base space. So if the base space is m-dimensional and the bundle group is n-dimensional, then μ ranges to m and r ranges to n. Here A can be any functions (locally, at least).

vector bundle

on a vector bundle, given a local trivialisation, a connection is described by a exterior covariant derivative, which is an element of End(V)⊗Ω(M) which satisfies the Leibniz law, so in local coordinates, you have

$Dv=(\partial _{\mu }v^{b}+v^{a}\Gamma _{\mu a}{}^{b})e_{b}\otimes dx^{\mu }$

where v is a section of the vector bundle. So if the vector space is k-dimensional, then a and b range up to k (these are End(V) indices). Here Γ can be any functions (at least locally).

associated bundle

if the vector bundle is the ρ-associated bundle to the principle bundle, then ρ takes elements of g to End(V) (End(V) is the same thing as Hom(V,V), but distinct from Aut(V). End(V) is the Lie algebra of Aut(V). if ρ is a faithful rep, it will actually map G to a subgroup of Aut(V) and g to a Lie subalgebra of End(V). So ρ(T) is actually an element of the image of the Lie algebra of g in End(V)). Thus we may write

$\Gamma _{\mu a}{}^{b}=\rho (T^{r})^{b}{}_{a}A_{r\mu }\;$

and so you can see how the covariant derivative on a vector bundle associated to a principle bundle explicitly depends on the connection form of the principle bundle. Here, a and b are k-dimensional ρ rep of g indices.

Lethe | Talk 22:17, 20 October 2005 (UTC)

Thanks Lethe, that is pretty clear. Could you also explain why A can represent the connection? --MarSch 10:45, 21 October 2005 (UTC)

Well, the connection form on the total manifold determines a connection by decreeing that the kernel of the form are "horizontal" spaces. A path is parallel transported if its tangent vector remains horizontal. The local connection form is just the pullback of the connection form by a local trivialisation. Most of my textbooks define a connection to be a splitting of the tangent space of the total space into horizontal and vertical components, so once you see how you can use a one form to split the tangent space into parts, you see why it can represent a connection. Some textbooks simply define the connection one form to be the connection, in which case it is called the Ehresmann connection. -Lethe | Talk 17:02, 21 October 2005 (UTC)

It does not seem like the connection form A depends on the local trivialization. This probably has to do with the isomorphism between the tangent bundle of a Lie group and the product of that Lie group with its Lie algebra. See also the discussion on this at http://en.wikipedia.org/wiki/Talk:Gauge_theory#Frame-dependent_claims --MarSch 14:41, 24 May 2006 (UTC)

Check my work?

I just added a section called "relation to Lie groups". I was unsure about some of the points, I only thought they were true. Irresponsibly, I added them anyway. I welcome comments and corrections.

1. the functor is full, faithful, and exact. Does the fact that the canonical projection SU(2) --> SO(3) goes to the identity prevent the functor from being faithful? I'm not sure. I'm positive it's full, but as for exactness, I feel like it should hold, but can't convince myself why (or why not).
2. The connected simply connected Lie group associated to a Lie algbra is isomorphic to the group of units of the completion of the universal enveloping algebra. I know that it can be embedded in this completion, but I don't recall hearing that it is the group of units. I just thought it ought to be.
3. Umm... I guess that's it. The rest is OK (I hope).

thanks -13:29, 30 October 2005 (UTC)

Re: the completion of the universal enveloping algebra. I'm at the limits of what I know here (I'm travelling so I'm without my favorite reference which is Hochschild's book) but you need to specify a topology(correction: uniform structure --CSTAR) for this to make sense. The enveloping algebra is a graded algebra so has an ultrametric, but this obviously won't do since ultrametric topologies are totally disconnected. The Enveloping algebra is an algebra of differential operators so maybe you can assign a topology in that way. But in any case, somebody more knowledgeable better comment on your completion remarks. Maybe Serre's ancient notes deal with this. --CSTAR 16:19, 30 October 2005 (UTC)

I'm not familiar with Hochschild, but I think I read about this in Barut. Which I don't have access to presently. Anyway, now I'm thinking this can't be right: if x is a unit, so is 2x. In general, they won't both be in the Lie group. I think there ought to be a contruction where you arrive at the Lie group from the enveloping algebra, but until I can find something in the literature, I should probably remove the remark. Anyway, regarding a topology for the enveloping algebra, can't we somehow get one from the Lie algebra? Like uh... well I don't know how. Nevermind. -Lethe | Talk 16:41, 30 October 2005 (UTC)

Completely reducible gl_n modules

I'm interested in learning more about complete reducibility.. Can anyone help me find a completely reducible gl_n module? This question is possibly lame..

New To Advanced Math

Hi; I'm trying desperately to understand many of these advanced principals of mathematics, such as Lie algebras, but no matter how many times I review the material, it doesn't sink in. Could someone please provide examples, problems to solve (with their solutions) and/or ways to visualize this? beno 26 Jan 2006

You posted the same comments at Talk:Lattice (group), and received a reply there. I can only suggest that whenever you find a term that you don't know, click on the link until you reach an article that you do understand. Then work forwards again, and backwards, as necessary. Google can help as well. linas 04:18, 1 February 2006 (UTC)

This guy posted the same exact message (with the corresponding article title swapped in) at about 20 math pages. I've replied to 2 or 3 of them, as have a few others. What's the deal with this guy? -lethe ^{talk +} 05:25, 1 February 2006 (UTC)

The link to Weyl's theorem

The link to Weyl's theorem currently leads to a disambiguation page. Someone with more knowledge of the subject should correct the link to lead to a specific article.--Bill 19:56, 3 February 2006 (UTC)

I've always known this as the unitary trick, so I've changed the link accordingly. -lethe ^{talk +} 20:52, 3 February 2006 (UTC)

Structure and classification

I've corrected multiple inaccuracies in that section (and renamed it). Since Lie theory part of Wikipedia has developed quite a bit in recent months, fairly large clean-up of this article may be in order (there are also internal redundancies). In particular, I can envisage moving out the structure theory to a separate article, with only a summary on this page. Arcfrk 05:58, 15 May 2007 (UTC)

Please remove the label ^{clarification needed}. The formulation here is correct, since there is no classification of solvable Lie algebras yet. — Preceding unsigned comment added by 130.133.155.70 (talk) 09:36, 14 October 2012 (UTC)

Do the Lie algebras form an abelian category?

No.

In fact, the morphisms between two Lie algebras don't form a abelian group, so the category of Lie algebras is not even an additive category. —Preceding unsigned comment added by 59.66.121.105 (talk) 15:03, July 9, 2007

But modules over a fixed Lie algebra do form an abelian category, in analogy with the associative case (Rings is non-abelian; R-mod for a fixed R is abelian). Arcfrk 16:30, 9 July 2007 (UTC)

It is stated every Lie Alegra has a representation via matrices which I guess correspond to multiplications of the elements of the matices but the lie group corresponding to the algebra is not unique as previously stated so how is this possible.If this can be simply explained . Is it a gobal versus a local thing and is a lie group only a local correspondence

Infinite dimensions and correspondence with "Lie" groups

I'd like a reference to the statement "some infinite-dimensional Lie algebras are not the Lie algebra of any group". I guess this holds for a specific definition of an infinite-dimensional Lie group, otherwise it would be quite remarkable to me. 147.122.53.27 (talk) 18:32, 12 May 2008 (UTC)

Nilpotency and the Killing form

I deleted the claim that nilpotency is equivalent to the vanishing of the Klling form as it is false: there are non-nilpotent Lie Algebras whose Killing form vanishes. Also, standard usage of "Cartan's criterion" refers to solvable or semisimple Lie algebras, not nilpotent ones. I discuss this in detail in Talk:Cartan's_criterion. Jeremy Henty (talk) 22:13, 20 June 2008 (UTC)

Generators and dimension

I feel like it would be a good idea to include definitions of "Generators of Lie Algebras" and "Dimension of a Lie Algebra", since both of these concepts are mentioned in the current version and are not defined.

I don't have any references handy or I would do it myself.

24.141.149.178 (talk) 23:39, 18 July 2008 (UTC)

It is a little hard to see where to put them. The dimension of a Lie algebra is the dimension as a vector space. An algebra over a field k is defined to be a k-vector space with additional structure; it retains its structure as a vector space, so has a dimension. For generators, this is just a subset not contained in any proper Lie subalgebra. For both, the definitions are so standard, they should just be defined casually, but on the other hand, I didn't see where to put them without interrupting. JackSchmidt (talk) 03:59, 19 July 2008 (UTC)

A casual definition would be fine. It would just be so that people new to the subject of Lie algebras won't be left wondering what the examples are talking about. Would it be possible to throw them in with the examples as they are mentioned? 129.97.91.130 (talk) 21:00, 23 July 2008 (UTC)

Actually, now that I've re-read the article, it would probably fit better in the Definition and First Properties section.129.97.91.130 (talk) 21:03, 23 July 2008 (UTC)

These still need to go in the article. There should be a link to Structure constants as well. Count Truthstein (talk) 01:45, 15 January 2013 (UTC)

A link

I am wondering whether it would be appropriate to include a link to liealgebras yahoo group http://tech.groups.yahoo.com/group/liealgebras/ Justpasha (talk) 18:22, 25 August 2008 (UTC)

Generally not see WP:LINKSTOAVOID. --Salix alba (talk) 18:45, 25 August 2008 (UTC)

Lie algebra, lol

"Lie algebra"? I thought that all algebra was "lies". e.g. This S is a 14 etc. :-P 74.209.54.156 (talk) 06:06, 18 July 2011 (UTC)

Symbols of Lie algebrae

Symbols of Lie groups, such as SO, U, Spin, certainly are not italicized. What experts say about Lie algebrae indeed? Even Orthogonal group #Lie algebra has an intermixed use of Roman and Italic lowercase "so" letters. Incnis Mrsi (talk) 15:43, 9 August 2012 (UTC)

Relationship to Lie groups

I think the article should describe the relationship to Lie groups more explicitly. It could have examples of what the Lie algebras are for some Lie groups (GL(n), SL(n), etc.), including how the Lie bracket is defined. The idea of members of the Lie algebra being infinitesimal generators for the Lie group through the exponential map is quite important, and could be explained more explicitly for some concrete examples of Lie groups. Count Truthstein (talk) 21:45, 15 December 2012 (UTC)

Lie algebra #Definitions and Count Truthstein

A good content, but one should not bloat the section. It is the time to move exponentiation to Lie algebra #Relation to Lie groups, an already existing section. Incnis Mrsi (talk) 15:48, 26 January 2013 (UTC)

I'm trying to think exactly how it should be moved. Some of the simple definitions could go in the definitions section, but then I feel that I want to explain the importance of these definitions. Anyway, I'll try to work something out. Count Truthstein (talk) 16:26, 26 January 2013 (UTC)

Changes by Kmarinas86

[1] unlikely contains an obvious improvement, but does not contain a single edit summary (except automatic ones from editing sections). Changes in the lead are contestable, and moving the paragraph about "fraktur" to the section about Lie groups was not explained. Should these changes by Kmarinas86 (talk · contribs) be reverted? Incnis Mrsi (talk) 10:00, 21 February 2013 (UTC)

The relationship between Lie brackets and Lie algebras should be explained, and more mention needs to be made of the Jacobi Identity. Also, there needs to be more continuity between statements in the article, particularly those in the same section. The section previously titled "Definitions" did not clearly indicate what plurality of definitions were being discussed, and it appeared to be a list of properties rather than an explanation for the functions which used were used as examples of these properties - the latter being more useful to the reader.siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 14:27, 21 February 2013 (UTC)

Problems under == Definitions == are not a good pretext to drop the section header at all. Also, notice multiple stylistic degradations (“Consider blah-blah” instead of an impersonal forms), and I repeat:

“ it is customary to differentiate a Lie algebra from a Lie group by using lower-case fraktur notation ”

is awful and may not start the section Lie algebra #Relation to Lie groups. The use of fraktur for (Lie) algebras is one point. The use of same letters (up to the case) for group symbols is another point. The article should, first, declare the use of fraktur, and only after than explain that same capitalized letters are used for group symbols. Incnis Mrsi (talk) 15:17, 21 February 2013 (UTC)

It might be helpful to have multiple definitions of Lie algebra if one wants to have a section titled "Definitions" in its article. If the "fraktur" is that important as you suggest, then we need a section titled "Notation" or perhaps as a subsection of the section "Definitions", assuming that more than one will be presented. Sincerely,siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 15:30, 21 February 2013 (UTC)

The changes don't appear to be an improvement. There is no definition of a Lie algebra any more. It's unclear what the section "Relationship with Lie brackets" is supposed to be about. Count Truthstein (talk) 19:17, 21 February 2013 (UTC)

After the revert, new changes have been made. They are a little too numerous to make a detailed assessment, but the overall sense is negative. The changes in the subsection headings and reordering of the material do not appear to be an improvement at all, with a possible exception of organizing the section on terminology by creating subheadings for Lie subalgebras, etc. I especially disapprove of moving the examples of Lie algebras to the end of the article. May I propose to revert back to the last stable version and then discuss any further changes on the talk page first and build a consensus? Arcfrk (talk) 21:35, 27 February 2013 (UTC)

The examples of Lie alegbras you mentioned are specifically those which were associated with Lie groups, which were already located inside section "Relation to Lie groups" (http://en.wikipedia.org/w/index.php?title=Lie_algebra&oldid=539832759#Relation_to_Lie_groups). It is quite unclear why moving these examples to the end of the section under a new subheading is a bad move. The current position I placed this at is the last subheading of the section "Lie groups" (http://en.wikipedia.org/w/index.php?title=Lie_algebra&oldid=539846529#Lie_groups_2). It actually makes more sense to describe the relationship to Lie groups before pulling examples out of nowhere (from the perspective of someone trying to learn the subject for the first time) without adequate explanation.siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 22:09, 27 February 2013 (UTC)

By the way, the old "Examples" section wasn't moved. It (http://en.wikipedia.org/w/index.php?title=Lie_algebra&oldid=539832759#Examples) was renamed and sectioned (http://en.wikipedia.org/w/index.php?title=Lie_algebra&oldid=539846529#Possible_sources). The new "Examples" subsection sits under the "Lie groups" section and was derived from content from that same section.siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk}

I agree that the overall sense of the changes is negative. I support a revert and we should discuss smaller changes. I don't see how "Possible sources" is a better section heading than "Examples". The "Lie groups" section has examples of Lie group to Lie algebra correspondences at the end, and you might think that you needed to understand all the category theory before it to benefit from it. I think it's better to put examples first. Also the "classification" section is harder to find - it's under the vaguely titled "Analogy to other group classifications" section. One improvement is the use of more subsections, which makes it easier to browse through the article to find what you're looking for. Looking at the table of contents, you can see that the structure of the article is worse. Count Truthstein (talk) 15:02, 4 March 2013 (UTC)

"I think it's better to put examples first." That's a very bad idea. "Exercise" and "Problem" sections are usually located at the back of chapter or subchapter, not introduced at the beginning. "Examples" are in some respects are similar to "Exercises" and "Problems", and in an encyclopedia, although there are not exercises and problems, technical examples most certainly do not belong within (or precedent to) paragraphs which set the contextual basis for them.

"Looking at the table of contents, you can see that the structure of the article is worse." You're not seeing the problem with having a lack of subsections in the sections of the previous version of the article. From a perspective from someone who is not familiar with the content, it is better to divide the sections into smaller subsections. I understand that someone who is already an expert at this material will see it differently, because usually the advanced are able to read through large technical sections without getting lost.

"Also the 'classification' section is harder to find - it's under the vaguely titled "Analogy to other group classifications" section." I can only agree with this if a "classification" section is to be expected in the article as much as an "Introduction", "Definition", or "History" section would be expected. However, I do not see that to be the case with a "Classification" section (save if this was an article about a particular taxon). Most readers are not looking for a "Classification" section, so I must disagree with this as well.siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 03:04, 5 March 2013 (UTC)

By the way, right under the heading "Structure theory and classification", there is a sentence that makes it clear that there is a multiple of classifications within the classification scheme. "The classification of Lie groups is related to the classification of real and complex Lie algebras." In other words, the classification of <many> is related to the classification of <many>. Before my edits, the article used to have a subsection titled "Classification", where it jumps ahead, talking about the "Levi decomposition" and skips all the other important bits (or should I say key examples of the classification of Lie algebras). All the subsections in this section clearly are related to classification of Lie algebras, yet prior to my edits, the last subsection was allegedly the section on the "Classification" of Lie algebras.siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 03:39, 5 March 2013 (UTC)

You don't seem to know what classification means. It is not the same as listing sub-types (Abelian...) of the object. Count Truthstein (talk) 15:28, 6 March 2013 (UTC)

"I don't see how "Possible sources" is a better section heading than "Examples"" Here's why it is better. Every example in that section deals with various mathematical structures from which a Lie algebra can be derived. These include: Vector spaces, Subspaces, Lie groups, Three dimensional space, and Infinite dimensions. It's troubling that the value of this has been overlooked.siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 03:27, 5 March 2013 (UTC)

Any mathematical definition defines an extension of mathematical objects. The reader will automatically know what an "Example" is because it's common to talk about examples of mathematical objects. "Possible sources" is idiosyncratic and could mean anything. Count Truthstein (talk) 15:23, 6 March 2013 (UTC)

Okay. By the way, thanks for looking through and filtering in the good parts of my edits. I was beginning to think that I wasted several hours for nothing - lol. I think I am done editing this article.siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 15:53, 6 March 2013 (UTC)

Replaced a section

(http://en.wikipedia.org/w/index.php?title=Lie_algebra&diff=542199733&oldid=542198701) The fact that word "manifold" only appeared once in the body, even though it appears in the introduction, seems rather bizarre to me. You have to admit that the previous section (http://en.wikipedia.org/w/index.php?title=Lie_algebra&oldid=537799143#Relation_to_Lie_groups) that I replaced completely lacks structure; it is not clear how one point leads to another. In contrast, the section I found on the article Lie groups was a breath of fresh air. Granted, some information was lost along the way, but if it is going to return, it has to be as well thought out as the imported section. It shouldn't return as the box of randomly assorted facts that it was.siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 13:08, 5 March 2013 (UTC)

I've reverted the changes noting the above objections. I also have an objection of my own. You are breaking the terms of the GFDL and the CC by SA by copying and pasting other articles into this one without attribution (and thus passing it off as your own). Don't do it. You need to justify this duplication also. IRWolfie- (talk) 21:27, 5 March 2013 (UTC)

In what sense does my copy and paste qualify as "passing it off as my own"? Also, I never heard of this problem that you speak of. Is it reasonable? Should I actually go through the edit history and list all the users who were involved in writing the text and put them in the edit summary? That sounds completely new to me. The obvious result of the consensus notwithstanding, I could not help but note how many misunderstandings were involved in these objections.siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 05:17, 6 March 2013 (UTC)

For good reference: http://en.wikipedia.org/w/index.php?title=Lie_algebra&oldid=542206098 (My most recent edit on this article as of today). The reverted version with the edit summary, "(Revert to earlier version. The new version seems to be change for the sake of it [sic] (and about an excess of sections))".siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 05:22, 6 March 2013 (UTC)

Oh goody. It turns out that I quoted an edit summary without proper attribution (where, presumably, I should have said who the author of the edit was). Asking this sounds a bit stupid, but "Did I violate the rule again?"siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 07:04, 6 March 2013 (UTC)

I find it hard to imagine that I would complain about someone copying, pasting, or moving material that I added to Wikipedia. On what higher standing should someone else have a problem with someone else copying, pasting, or moving material without attribution when they have chose to put it on Wikipedia article space of all things? I thought this rule applied only to files, user space, talk pages, etc. where text is easily regarded as having one identifiable author (such as the text you are reading right now), and not paragraphs in article space which are possibly written by 10 or more different authors. Geee. Is there not a difference to discern here?

http://en.wikipedia.org/wiki/Wikipedia:General_disclaimer

“ There is no agreement or understanding between you and Wikipedia regarding your use or modification of this information beyond the Creative Commons Attribution-Sharealike 3.0 Unported License (CC-BY-SA) and the GNU Free Documentation License (GFDL); neither is anyone at Wikipedia responsible should someone change, edit, modify or remove any information that you may post on Wikipedia or any of its associated projects. ”

siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 05:36, 6 March 2013 (UTC)

"You need to justify this duplication also." It could have been "altered" so as to no longer remain a duplicate. Until Wikipedia data structure becomes more like an Access database where the visitor can select the mode of presentation (e.g. chronological, alphabetical, subject matter, difficulty, etc.) for the same data set, there will no doubt be cases where similar material and sometimes duplicate material will appear in multiple contexts in which such material is relevant.siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 06:00, 6 March 2013 (UTC)

For the record:

My version of the section on the relationship between Lie groups and Lie algebras. (http://en.wikipedia.org/w/index.php?title=Lie_algebra&oldid=542206098#The_Lie_algebra_associated_with_a_Lie_group) And, if it's not clear enough, "My version" does not mean "My writings". Thanks.

The "preferred" version. (http://en.wikipedia.org/w/index.php?title=Lie_algebra&oldid=542264704#Relation_to_Lie_groups)

siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 06:00, 6 March 2013 (UTC)

Stop rambling. You can not copy and paste other articles without saying where you got it. You just can't and it's in the policies I put on your user page. IRWolfie- (talk) 10:11, 6 March 2013 (UTC)

I would find it most reasonable if the request were that (i.e. saying where I got it from). You seemed to be saying that I had to state WHO I got it from (what "attribution" implies). Well in that case, it shouldn't be too hard to state where I got them from:

Sources for the copied and pasted material (oldid=542206098):

• Lie Algebra

• Lie group#The Lie algebra associated with a Lie group

• Properties

• Covering group#Lie groups

• Exponential Map

• Exponential map (Lie theory)

• Types of Lie groups and their associated Lie alegbras

• Lie group#The concept of a Lie group, and possibilities of classification
• Lie group#Types of Lie groups and structure theory

• Informal definition & General definition

• Lie group#The Lie algebra associated with a Lie group

Note: All the copy-and-paste material regarding this incident are all focused on the same section.siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 13:53, 6 March 2013 (UTC)

You should not be making further drastic edits to the article when the changes you have already made are disputed. Count Truthstein (talk) 15:05, 6 March 2013 (UTC)

Ok.siNkarma86—Expert Sectioneer of Wikipedia
^{86 = 19+9+14 + karma = 19+9+14 + talk} 15:12, 6 March 2013 (UTC)

Relation to Lie groups

The section looks nice. But shouldn't this material be in Lie group? A Lie algebra is an algebra structure after all. -- Taku (talk)

I'd say that most of the material is relevant to both articles. YohanN7 (talk) 18:02, 6 June 2015 (UTC)

I agree. I was wondering (long time ago) whether this is the best place for such materials. Lie algebras can be studied for their own sake; just as an algebraic structure. It is more economical to concentrate the same materials at the same place; easier to maintain and expand. -- Taku (talk) 21:09, 6 June 2015 (UTC)