Talk:Universal enveloping algebra

message: 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)
replies: text: 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)
thread_title: alternate construction 

message: 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)
replies: text: I think this problem is now solved. 67.198.37.16 (talk) 21:01, 20 September 2016 (UTC)
thread_title: Direct construction 

message: 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:
* 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. 
67.198.37.16 (talk) 21:21, 22 September 2016 (UTC)
thread_title: TODO List

message: 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)
replies: text: 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)
thread_title: Mistake regarding the generalized Casimir elements?

message: 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<xref> Section 9.3</ref> by taking the quotient by imposing the relations
:a \otimes b - b \otimes a = [a,b]
for all  and  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)
replies: text: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Profoundly confusing explanation 

message: 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)
thread_title:  Linear combinations over what coefficients? 

message: 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)
thread_title:  Article would greatly benefit from a rewriting 

message: 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)
thread_title:  First sentence is confusing