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Recall the following classical theorem of Cartan (!):

Theorem (Lie III): Any finite-dimensional Lie algebra over $\mathbb R$ is the Lie algebra of some analytic Lie group.

Similarly, one can propose "Lie III" statements for Lie algebras over other fields, for super Lie algebras, for Lie algebroids, etc.

The proof I know of the classical Lie III is very difficult: it requires most of the structure theory of Lie algebras.

But why should it be difficult? For example, for finite-dimensional Lie algebra $\mathfrak g$ over $\mathbb R$, the Baker-Campbell-Hausdorff formula (the power series given by $B(x,y) = \log(\exp x \exp y)$ in noncommuting variables $x,y$; it can be written with only the Lie bracket, no multiplication) converges in an open neighborhood of the origin, and so defines a unital associative partial group operation on (an open neighborhood in) $\mathfrak g$. What happens if one were to try to simply glue together copies of this open neighborhood?

Alternately, are there natural variations of Lie III that are so badly false that any easy proof of Lie III is bound to fail?

flag	asked Dec 13 at 19:31 Theo Johnson-Freyd 6,118●4●32

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For the record, the modern statement of Ado's Theorem is a bit stronger. It says that a finite-dimensional Lie algebra is (isomorphic to) a matrix Lie algebra, and moreover such an isomorphism can be chosen so that the nilpotent part of the Lie algbera consists of nilpotent matrices. Ado's Theorem also seems to require most of the structure theory of Lie algebras. My standard reference for Lie theory are my notes from Prof. Haiman's class last year: math.berkeley.edu/~theojf/LieGroupsBook.pdf – Theo Johnson-Freyd Dec 13 at 19:37

Great notes Theo, thanks! – Grétar Amazeen Dec 13 at 20:00

Mariano Suárez-Alvarez · Accepted Answer · 2009-12-13 19:46:37Z UTC

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That gluing together of group chunks, constructed from the BCH formula is ~~precisely~~ more or less what Serre does to prove the theorem (in the first proof he gives in) his book on Lie groups and Lie algebras. [Serre, Jean-Pierre. Lie algebras and Lie groups. 1964 lectures given at Harvard University. Second edition. Lecture Notes in Mathematics, 1500. Springer-Verlag, Berlin, 1992. viii+168 pp.]

edited Dec 13 at 19:46

answered Dec 13 at 19:41

Mariano Suárez-Alvarez
10.5k●17●55

I agree with MSA. Look in Serre's Lie Algebras and Lie Groups: he proves Lie III for Lie algebras over any nondiscrete, locally compact field. Part I of the book is on Lie algebras, but it is generally agreed that it covers only the more formal aspects and not the fine structure theory of Lie algebras: e.g. no roots or weights. (His book Complex Semisimple Lie Algebras has more of this sort of thing.) Accordingly, I am no expert but I would say that Lie III is in fact not that difficult compared to some other big theorems in basic Lie theory. – Pete L. Clark Dec 13 at 19:56

David Speyer · Answer 2 · 2009-12-13 19:33:37Z UTC

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There are Lie algebras which are not the Lie algebra of an algebraic group. (See here.) That rules out some proofs, but it doesn't seem like it would be a problem for the sort of analytic proof you are proposing.

answered Dec 13 at 19:33

David Speyer
15.6k●1●11●57

Why is Lie’s Third Theorem difficult?

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