Talk:Universal enveloping algebra

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)

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)

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:

• 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)

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 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)