Talk:Universal enveloping algebra

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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)[reply]

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)[reply]

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)[reply]

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)[reply]

I think this problem is now solved. 67.198.37.16 (talk) 21:01, 20 September 2016 (UTC)[reply]

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)[reply]

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)[reply]

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. In particular, 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. Need short discussions of:

67.198.37.16 (talk) 21:21, 22 September 2016 (UTC)[reply]