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(One can of course define Lie algebras over other fields than the complex numbers , but in order to avoid some technical issues we shall work solely with the complex case in this post.)
An important special case of the abstract Lie algebras are the concrete Lie algebras, in which is a vector space of linear transformations on a vector space (which again can be either finite or infinite dimensional), and the bilinear form is given by the usual Lie bracket
It is easy to verify that every concrete Lie algebra is an abstract Lie algebra. In the converse direction, we have
To prove this theorem, we introduce the useful algebraic tool of the universal enveloping algebra of the abstract Lie algebra . This is the free (associative, complex) algebra generated by (viewed as a complex vector space), subject to the constraints
This algebra is described by the Poincaré-Birkhoff-Witt theorem, which asserts that given an ordered basis of as a vector space, that a basis of is given by “monomials” of the form
where is a natural number, the are an increasing sequence of indices in , and the are positive integers. Indeed, given two such monomials, one can express their product as a finite linear combination of further monomials of the form (3) after repeatedly applying (2) (which we rewrite as ) to reorder the terms in this product modulo lower order terms until one all monomials have their indices in the required increasing order. It is then a routine exercise in basic abstract algebra (using all the axioms of an abstract Lie algebra) to verify that this is multiplication rule on monomials does indeed define a complex associative algebra which has the universal properties required of the universal enveloping algebra.
The abstract Lie algebra acts on its universal enveloping algebra by left-multiplication: , thus giving a map from to . It is easy to verify that this map is a Lie algebra homomorphism (so this is indeed an action (or representation) of the Lie algebra), and this action is clearly faithful (i.e. the map from to is injective), since each element of maps the identity element of to a different element of , namely . Thus is isomorphic to its image in , proving Theorem 1.
In the converse direction, every representation of a Lie algebra “factors through” the universal enveloping algebra, in that it extends to an algebra homomorphism from to , which by abuse of notation we shall also call .
One drawback of Theorem 1 is that the space that the concrete Lie algebra acts on will almost always be infinite-dimensional, even when the original Lie algebra is finite-dimensional. However, there is a useful theorem of Ado that rectifies this:
Theorem 2 (Ado’s theorem) Every finite-dimensional abstract Lie algebra is isomorphic to a concrete Lie algebra over a finite-dimensional vector space .
Among other things, this theorem can be used (in conjunction with the Baker-Campbell-Hausdorff formula) to show that every abstract (finite-dimensional) Lie group (or abstract local Lie group) is locally isomorphic to a linear group. (It is well-known, though, that abstract Lie groups are not necessarily globally isomorphic to a linear group, but we will not discuss these global obstructions here.)
Ado’s theorem is surprisingly tricky to prove in general, but some special cases are easy. For instance, one can try using the adjoint representation of on itself, defined by the action ; the Jacobi identity (1) ensures that this indeed a representation of . The kernel of this representation is the centre . This already gives Ado’s theorem in the case when is semisimple, in which case the center is trivial.
The adjoint representation does not suffice, by itself, to prove Ado’s theorem in the non-semisimple case. However, it does provide an important reduction in the proof, namely it reduces matters to showing that every finite-dimensional Lie algebra has a finite-dimensional representation which is faithful on the centre . Indeed, if one has such a representation, one can then take the direct sum of that representation with the adjoint representation to obtain a new finite-dimensional representation which is now faithful on all of , which then gives Ado’s theorem for .
It remains to find a finite-dimensional representation of which is faithful on the centre . In the case when is abelian, so that the centre is all of , this is again easy, because then acts faithfully on by the infinitesimal shear maps . In matrix form, this representation identifies each in this abelian Lie algebra with an “upper-triangular” matrix:
This construction gives a faithful finite-dimensional representation of the centre of any finite-dimensional Lie algebra. The standard proof of Ado’s theorem (which I believe dates back to work of Harish-Chandra) then proceeds by gradually “extending” this representation of the centre to larger and larger sub-algebras of , while preserving the finite-dimensionality of the representation and the faithfulness on , until one obtains a representation on the entire Lie algebra with the required properties. (For technical inductive reasons, one also needs to carry along an additional property of the representation, namely that it maps the nilradical to nilpotent elements, but we will discuss this technicality later.)
This procedure is a little tricky to execute in general, but becomes simpler in the nilpotent case, in which the lower central series becomes trivial for sufficiently large :
Theorem 3 (Ado’s theorem for nilpotent Lie algebras) Let be a finite-dimensional nilpotent Lie algebra. Then there exists a finite-dimensional faithful representation of . Furthermore, there exists a natural number such that , i.e. one has for all .
The second conclusion of Ado’s theorem here is useful for induction purposes. (By Engel’s theorem, this conclusion is also equivalent to the assertion that every element of is nilpotent, but we can prove Theorem 3 without explicitly invoking Engel’s theorem.)
Below the fold, I give a proof of Theorem 3, and then extend the argument to cover the full strength of Ado’s theorem. This is not a new argument – indeed, I am basing this particular presentation from the one in Fulton and Harris – but it was an instructive exercise for me to try to extract the proof of Ado’s theorem from the more general structural theory of Lie algebras (e.g. Engel’s theorem, Lie’s theorem, Levi decomposition, etc.) in which the result is usually placed. (However, the proof I know of still needs Engel’s theorem to establish the solvable case, and the Levi decomposition to then establish the general case.)