Comments on: Ado’s theorem

By: Notes on the classification of complex Lie algebras | What's new

Notes on the classification of complex Lie algebras | What's new — Sun, 28 Apr 2013 05:25:10 +0000

[...] to be a homomorphism into a concrete Lie algebra . It is a deep theorem of Ado (discussed in this previous post) that every abstract Lie algebra is in fact isomorphic to a concrete one (or equivalently, that [...]

By: Ado’s theorem for groups with dilations? « chorasimilarity

Ado’s theorem for groups with dilations? « chorasimilarity — Fri, 21 Sep 2012 11:28:44 +0000

[...] proof I am aware of, (see this post for one proof and relevant links), mixes the following [...]

By: Terence Tao

Terence Tao — Thu, 05 Apr 2012 15:18:32 +0000

The restriction of the action of on to is simply the original action of on , as can be seen by substituting into (5). The behaviour of becomes irrelevant at that point.

By: Theo

Theo — Thu, 05 Apr 2012 12:02:09 +0000

Just above Remark 1; why is the enlarged action still faithful on \mathfrak{a}? Couldn’t an element of \mathfrak{h} sit in the centralizer of \mathfrak{a} in \mathfrak{n}?

thanks

By: Associativity of the Baker-Campbell-Hausdorff formula « What’s new

Associativity of the Baker-Campbell-Hausdorff formula « What’s new — Sat, 29 Oct 2011 19:25:31 +0000

[...] With the assistance of Ado’s theorem, which places inside the general linear Lie algebra for some , one can deduce both the well-definedness and associativity of (3) from the Baker-Campbell-Hausdorff formula for . However, Ado’s theorem is rather difficult to prove (see for instance this previous blog post for a proof), and it is natural to ask whether there is a way to establish these facts without Ado’s theorem. [...]

By: 254A, Notes 1: Lie groups, Lie algebras, and the Baker-Campbell-Hausdorff formula « What’s new

Fri, 02 Sep 2011 02:15:30 +0000

[...] a global Lie group, requiring the non-trivial algebraic tool of Ado’s theorem (discussed in this previous blog post); see Exercise 20 [...]

By: Notes on local groups « What’s new

Notes on local groups « What’s new — Thu, 18 Aug 2011 03:04:38 +0000

[...] this first for Parts 1 and 2. Let be a Lie algebra. Applying Ado’s theorem (discussed in this blog post), we can identify with a subalgebra of the Lie algebra for some finite . If is a small symmetric [...]

By: Wolfgang M.

Wolfgang M. — Tue, 14 Jun 2011 15:07:31 +0000

Some (historical) remarks which may be of interest to some:

(1.) The above proof, in which a representation of a subspace is inductively extended to the whole Lie algebra, indeed seems to be the ‘standard’ proof for Ado’s theorem. A slightly different approach, by Y. Neretin (“A CONSTRUCTION OF FINITE-DIMENSIONAL FAITHFUL
REPRESENTATION OF LIE ALGEBRA,” http://www.ams.org/mathscinet-getitem?mr=1982443), can be found in the literature.

The objects and techniques used in his proof are almost exactly the same but with the following difference: Neretin first embeds the (complex, finite-dimensional) Lie algebra into a Lie algebra that is the semidirect product of a reductive Lie algebra and a nilpotent ideal such that the acts completely reducibly on the . It then suffices to find a (finite-dimensional) faithful representation of . This is done exactly as in the proof above: by letting act on its universal enveloping algebra through derivations (p) and left multiplications (h), and then considering the quotient , where is an invariant ideal that is spanned by all elements that are long enough, in some sense.

The embedding theorem follows from what Neretin calls “elementary expansions,” based on the Jordan-Chevalley decomposition for derivations. I suppose this idea is also implicitly contained in the standard proof.

Neretin’s proof is short and elegant, but it also assumes theorem 7, corollary 8, theorem 10, and so on.

(2.) Concerning “Remark 2″: There have been attempts to determine (bounds for) the minimal degree of a faithful representation of a given Lie algebra in terms of other natural invariants associated to . Some classes admit polynomial bounds, but the general bounds appear to be exponential in the dimension.

By: Ravi

Ravi — Sat, 11 Jun 2011 12:15:19 +0000

I think there is a typo in (the statement of) Corollary 5. It should read “… $\text{tr}(X_1\ldots X_m)=0$” (not “… $\text{tr}(AX_1\ldots X_m)=0$”).

Ravi Raghunathan

[Corrected, thanks - T.]

By: Ramdorai

Ramdorai — Tue, 17 May 2011 20:17:34 +0000

Hello Vanessa. Such a Lie algebra as you are considering is called an m-Engel Lie algebra (this notion has been very widely studied by Efim Zelmanov in the case of infinite dimensions). I do not know a counterexample in the case of finite dimension.

In <> ( http://www.springerlink.com/content/j7562232386266w8/ ) your question appear as a conjecture.