Talk:Solvable Lie algebra

Importance?
Ridiculous nomination. Charles Matthews 19:14, 31 May 2007 (UTC)


 * It is incorrect to say that this page is totally contained in the Lie algebra page. The reference to Borel subalgebra isn't there. The links to the basic result, the Lie-Kolchin theorem, isn't there. The link to solvmanifold isn't there. The links to the EoM are not there. That's five items that are going to be useful to anyone who wants to know about solvable Lie algebras. Charles Matthews 19:26, 31 May 2007 (UTC)


 * And the definition of solvable Lie group? Where is that on the Lie algebra page? Charles Matthews 19:31, 31 May 2007 (UTC)

This is an extremely important topic in the theory of Lie algebras and it is important that it has a separate article. Geometry guy 00:46, 1 June 2007 (UTC)


 * Considering that every text on Lie algebras I've picked up over the years spends no more than a few pages on solvability, I think it would be pretty reasonable to make this a redirect. The solvable case is the throwaway case in the theory as it stands today. One could say that solvability is interesting in that it is the exact opposite of semisimplicity, which is interesting. Or perhaps that adjoining solvable one-dimensional extensions (read: central extensions) in view of Lie algebra cohomology is important. Either way, this page is very silly. If there are any bits of information on it that is not in the main article, they should be incorporated. Myrkkyhammas 09:25, 3 June 2007 (UTC)

That's not right at all. Google for "solvable Lie algebra" or "solvable Lie algebras" and you'll immediately see dozens of recent research papers on the solvable case, ranging from abstract algebra to physics. says there are two important cases (solvable, semisimple), not one. Don't go around saying pages are very silly, when that reflects more on you. (Why would the Springer Encyclopedia have an article, if your rather superficial reading of the literature was right?) Charles Matthews 11:47, 3 June 2007 (UTC)

This article is OK! But it should be enlarged by at least the decomposition of a Lie algebra into a (semi-direct) sum of a semi-simple Lie algebra (the Levi factor) and a solvable ideal (theorem of Ado-Iwasawa). So every non semi-simple Lie algebra gives rise to a solvable one. The structure theory of semi-simple Lie algebras is known - that of the solvable Lie algebras not! This should be mentioned for completeness and for pointing to an open problem in algebra. In addition there are examples of the type Heisenberg Lie algebra (nilpotent) of a symplectic vector space + a Hamiltonian, giving rise to solvable Lie algebras of dimension 2n+2. — Preceding unsigned comment added by 134.60.206.14 (talk) 10:48, 11 April 2013 (UTC)

Regarding the comment that Lie theory texts spend only a few pages on solvability: IMO this is not because solvable Lie algebras and groups are unimportant, but rather because classifying them or resolving them into readily described classes (as can be done for semisimple algebras) is so far too damn hard. The Killing form's nondegeneracy for semisimple groups is a big part of why these latter guys are so "easy" to study: the solvable algebras, have degenerate Killing forms, thus a nontrivial Kernel which is a murky world where the Killing form yields no more information. RodVance (talk) 04:31, 30 May 2015 (UTC)


 * For an important (= useful) solvable (nilpotent actually) Lie algebra, see Heisenberg algebra. Major examples of application of it and its representations are QFT and string theory. YohanN7 (talk) 11:29, 1 June 2015 (UTC)