Talk:Universal enveloping algebra

alternate construction
the first sentence of the section which i renamed "Alternate Construction", namely "Noting that any associative K-algebra becomes a Lie algebra with the bracket [a,b] = a.b-b.a, a construction and precise..." i added this sentence to some stuff i put at the top in some explanatory stuff. if this stuff stays at the top, then this sentence should be removed from the "alternate Construction" section, because it would be redundant. i didn't remove it myself, because i couldn't figure out how to start the paragraph without it. need help of the author (i assume this is you, Charles?) - Lethe

I've been through this again, mainly format matters, but some moves of material.

Charles Matthews 14:45, 13 May 2004 (UTC)

Charles, do we have the ability to make commutative diagrams?

There are some examples (Clifford algebra, IIRC correctly). Mostly from the old days, and not very nice. Or, people make little graphics to upload.

My taste is to use words, anyway. This really isn't a mathematics text, from the point of view of exposition.

Charles Matthews 21:24, 13 May 2004 (UTC)


 * OK then. another question: i put the universal construction on top of the less abstract construction, but looking around a bit (e.g. Tensor product) it seems that the preference is to start with the less abstract, and save the universal property for later.  what do you think?

Not a big deal, either way. In this case, relying on tensor algebra, it's kind of clear what to do. Charles Matthews 11:37, 14 May 2004 (UTC)

Direct construction
I changed the first sentence of "direct construction." It used to say "For general reasons having to do with universal properties..." the thing is unique if it exists. I thought this wording was oblique at best and incorrect at worst. Most directly, one would probably prove uniqueness from the universal property via some standard abstract nonsense argument. There's probably some way to formulate things so that the proof is seen as following from a "general reason," but more likely I would say it follows from a "standard method." Either way, the present wording, hinting, imprecisely and mysteriously, at grandiose ideas doesn't seem helpful. Either stick with my approach (though please reword it-- I don't know if my wording is ideal) or describe the proof-process in a little more detail, if you're going to mention it. Lewallen 01:55, 12 March 2007 (UTC)
 * I think this problem is now solved. 67.198.37.16 (talk) 21:01, 20 September 2016 (UTC)

Example?
The most familiar nontrivial example of a Lie Algebra would be, I'd guess, the cross product.

It would be nice to see explain, by way of an example, what the universal enveloping algebra of the cross product is.

Just my 2c... mike40033 (talk) 00:25, 13 February 2014 (UTC)


 * I think I just now accomplished this, I hope. The article now gives a very highly detailed development of how, exactly, one builds the thing. All you need to do is to find every location of the Lie bracket in that section, and replace it by the cross product, and bingo, you're done. If perhaps, somehow, that is still not enough to understand the concept, then re-read the "intuitive definition" section.  If that is not enough, and you want a concrete, explicit example, then you must jump forward to the very end, and read the final section, which provides the key construction. Here is the semi-accurate, and very explicit, concrete description:


 * The Lie group corresponding to the cross product has a manifold, and that manifold is the 3-sphere (well, it depends on which Lie group exactly, but the 3-sphere covers them, so lets go with that). The universal covering algebra for the cross-product is then, more or less, the vector space of all continuous complex-valued functions on the 3-sphere. This is clearly a very big space! I would put this in the article, except that I don't know of (can't think of) an easy, "obvious" straight-forward proof of this - the blather about Hopf algebras at the end being non-obvious.  But it is a worthy undertaking to find the simplest, easiest proof of such a construction.  I can't say that I've ever seen such a thing ... have to think about this ... it would make a worth-while addition to this article.  Anyway thanks!  I've just learned something new.  67.198.37.16 (talk) 20:57, 20 September 2016 (UTC)

TODO List
I've more-or-less completely rewritten this article. I hope that it is relatively clear now. There do remain various topics that really need to be fleshed out. Need short discussions of: 67.198.37.16 (talk) 21:21, 22 September 2016 (UTC)
 * Milnor–Moore theorem
 * Harish-Chandra homomorphism
 * Expanded discussion of the representation theory, starting with Verma modules, which are most easily accessible from current content.
 * More correctly, PBW says that the envelope is the coordinate ring $$k[\mathfrak{g}^*]$$ with g-star the dual vector space, which follows from the free vector space used in the construction being covariant, instead of contravarient, the way ordinary vector spaces are. The distinction between co and contravarience were glossed over, and could be fixed. Its a subtle point that usually does not matter.
 * The interplay with commutative geometry needs to be made explicit and formal: the intro says that "it kind-of-ish looks like the space of functions on the group manifold", this needs to be made formal. You can already smell this with all that talk about derivations and polynomials, and the algebras of Lie derivatives on manifolds. However, its all still vague and should be made concrete.
 * Relationship to physics, which are multifold. Well-known are the deformations that lead to the quantum groups. An then the spin structures. Much more obscure is why it is that the formulas resemble the Maxwell-Boltzmann statistics, and how this generalizes to other spin-statistics.

Mistake regarding the generalized Casimir elements?
The article claims: "It is clear that all central elements will be linear combinations of symmetric homogenous polynomials in the basis elements e_{a} e_{a} of the Lie algebra.", but the Casimir element of sl(2,C) is scalar multiple of h^2+2ef+2fe, which is not a symmetric polynomial in the usual sense. Are they trying to say the image under the Harish-Chandara isomorphism is symmetric? Or is it symmetric with respect to some other group action? -Pabnau 04:04, 16 Feb 2018 (UTC)


 * I think you need to write it in the symmetric basis, instead of in the ladder operators. The PBW theorem asked you to to create a symmetric basis. The quadratic casimir for sl(2,C) is the same as for su(2) aka so(3) in the examples; su(2) is just the real part of sl(2,C). 67.198.37.16 (talk) 04:41, 18 October 2020 (UTC)

Profoundly confusing explanation
The technical definition of universal enveloping algebra reads as though it was written by someone who feels profoundly confused by the concept.

We need a definition written by someone who understands the concept well enough not to write a profoundly confusing definition.

To take just one example: From the section Formal definition we find:

The universal enveloping algebra is obtained Section 9.3 by taking the quotient by imposing the relations


 * $$a \otimes b - b \otimes a = [a,b]$$

''for all $a$ and $b$ in the embedding of $$\mathfrak{g}$$ in $$T(\mathfrak{g}).$$ To avoid the tautological feeling of this equation, keep in mind that the bracket on the right hand side of this equation is actually the abstract "bracket" operation on the Lie algebra. Recall that the bracket operation on a Lie algebra is any bilinear map of $$\mathfrak{g}\times\mathfrak{g}$$ to $$\mathfrak{g}$$ that is skew-symmetric and satisfies the Jacobi identity. This bracket is not necessarily computed as $$[X,Y]=XY-YX$$ for some associative product structure on $$\mathfrak{g}$$.''

But there is nothing in the definition that would distinguish one universal enveloping algebra from another.

I recognize that someone has attempted to clarify this in the text.

However: That is not what "formal definition" means. It does not mean an inadequate definition that is supplemented by a few English sentences in the hope they will neutralize the confusion sown by the inadequate definition.

(Maybe you though the brackets might be different in various universal enveloping algebras? A definition must make this explicit and not merely hope that readers will intuit this by telepathy.)2600:1700:E1C0:F340:1D1A:891C:4C07:FC9D (talk) 05:33, 6 November 2018 (UTC)


 * I just want to point out that we do give more formal definition after that. But, yes, the writing has a problem you identified and has to be fixed (by you, me or any other human or bot...). -- Taku (talk) 00:27, 12 April 2019 (UTC)


 * It would be nice if someone could explain in greater detail what is confusing. An earlier version contained a much longer explanation here, showing how to lift the Lie bracket on a vector space to a Lie bracket on the tensor algebra. It was cut and replaced by the above. Would an explanation of the lifting clarify things? (I liked the lifting approach, it was verbose but spelled out all the details.) I'll see if I can repair this. I just got done hacking Hall word and was going to integrate that article into this one, anyway. 67.198.37.16 (talk) 05:50, 17 October 2020 (UTC)


 * OK I think I just fixed it. I restored the text about lifting, so that is now explicitly spelled out. I've also re-organized the discussion of ideals. So now one has not one, but two equivalent definitions. You can bounce one off the other and see that they are consistent.67.198.37.16 (talk) 19:06, 17 October 2020 (UTC)

Universal Property
In the section on the universal property, it is said that
 * This is a functor Lie from the category of algebras Alg to the category of Lie algebras LieAlg over some underlying field—in fact, it is a free functor.

This just seems wrong, because there is no forgetful functor from LieAlg to Alg, or is there? Consequentley, there cannot be a left adjoint, i.e. a free functor.

The last sentence is
 * By functor composition, the universal enveloping algebra constructs an adjunction between the free Lie algebra on a vector space and the forgetful functor from Lie algebras to vector spaces.

which seems wrong as well. The adjunction "free Lie algebra from Vect" ⊣ "forget" is
 *  LieAlg(freeLie(V), g) = Vect(V, forget g),

the adjunction "universal enveloping algebra" ⊣ "free Lie algebra from Alg" is
 * Alg(U(g), A) = LieAlg(g, Lie(A)).

So, writing the above quote in formulas, we get
 * Vect(V, forget U (g)) = Alg(free V, U g) =/= LieAlg(freeLie V, g).

134.100.222.237 (talk) 08:01, 9 April 2019 (UTC)


 * I cannot figure out what you are saying. There is a forgetful functor from LieAlg to Vect. I think that's the first thing that section says. Just forget its an algebra, all you got left is the vector space. There is also something called the free Lie algebra: just take a vector space, generate and assoc alg A from it, and then impose an anti-commutator on A. This is freeLie(V) There are lots of canonical ways of writing the basis for this - I just got done writing the article on Hall word; it is one such basis, but so are the Lyndon words.) So we've got the needed hom's.


 * Using your notation, we have forget(g)=V and given V, we have freeLie(V) Next is the question of "what do you mean, when you write LieAlg(freeLie(V), g)"? You have two choices here: You can take this to be g or you can take it to be U(g).  Saying that its Vect(V, forget g) doesn't make sense, because V is already a Vect, and forget(g) is V. The intent is to say "when you impose g (i.e. the commutator relations) on freeLie(V) you get U(g)". That's because freeLie(V) already moded out anything anti-commuting, so there's nothing anti-commuting left over. All that's left is to mod out the relations in g.


 * I can't parse your last formulas. A is already an algebra, and so is U(g) and A has less structure than U(g) so you can't impose it on U(g), and besides, we are not forgetting a LieAlg to become an Alg, we are forgetting a LieAlg to become a V. I think the article is correct as it stands; its a bit hard to convert the words into formulas; I suppose this could be done but it would blow up the section and make it larger. Heh. 67.198.37.16 (talk) 22:22, 17 October 2020 (UTC)


 * Hmm. On closer reading, there's a number of awkwardly worded statements about T and U in there. It should say something like this: T is a functor that takes V to T(V) and not to FreeAlg... there is a functor that takes T(V) to FreeLie(V)= ... there is a functor Lie that takes Alg to LieAlg ... and takes FreeLie to U(g) ... Ugh.... this needs clarification. 67.198.37.16 (talk) 22:34, 17 October 2020 (UTC)

Linear combinations over what coefficients?
The section Free associative algebra begins with this passage:

"Recall first the definition of a free associative algebra generated by $$n$$ elements $$x_1,\ldots x_n.$$ This consists of all possible linear combinations of finite sequences $$x_i^mx_j^n\cdots x_k^p$$ for arbitrary (non-negative) integer powers $$m,n,\cdots,p$$ and the index drawn from the generating set, i.e. $$1\le i,j,\cdots,k\le n.$$ "

But linear combinations require coefficients belonging to some particular ring or field. This coefficient ring needs to be specified.65.132.0.175 (talk) 16:31, 17 November 2020 (UTC)

Article would greatly benefit from a rewriting
This article on one hand uses too many words, which in many places have an extremely unclear meaning.

By the same token, the article does not have enough words that are clear.

The article could greatly benefit from being rewritten so that each sentence follows logically from what came before, without all the extra verbiage that is in so many cases unclear. I hope someone knowledgeable in this subject can do this.65.132.0.175 (talk) 16:59, 17 November 2020 (UTC)

First sentence is confusing
The article starts:


 * In mathematics, a universal enveloping algebra is the most general (unital, associative) algebra that contains all representations of a Lie algebra.

First, I don't think it's true: if we take the Lie algebra to be the 1-dimensional abelian Lie algebra, its universal enveloping algebra does not contain all the representations of this Lie algebra (which has a continuum of 1-dimensional representations). Second, this is definitely not the point of the universal enveloping algebra. So I'm going to change this sentence to


 * In mathematics, the universal enveloping algebra of a Lie algebra is the associative unital algebra whose representations correspond precisely to those of that Lie algebra.

Maybe this can be improved without making it too long, but this is basic idea behind the universal enveloping algebra. Note also that it's important to talk about the universal enveloping algebra of a Lie algebra. John Baez (talk) 19:05, 10 January 2022 (UTC)