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We will usually omit the subscript from the Lie bracket when this will not cause ambiguity. A homomorphism between two Lie algebras is a linear map that respects the Lie bracket, thus for all . As with many other classes of mathematical objects, the class of Lie algebras together with their homomorphisms then form a category. One can of course also consider Lie algebras in infinite dimension or over other fields, but we will restrict attention throughout these notes to the finite-dimensional complex case. The trivial, zero-dimensional Lie algebra is denoted ; Lie algebras of positive dimension will be called non-trivial.
Lie algebras come up in many contexts in mathematics, in particular arising as the tangent space of complex Lie groups. It is thus very profitable to think of Lie algebras as being the infinitesimal component of a Lie group, and in particular almost all of the notation and concepts that are applicable to Lie groups (e.g. nilpotence, solvability, extensions, etc.) have infinitesimal counterparts in the category of Lie algebras (often with exactly the same terminology). See this previous blog post for more discussion about the connection between Lie algebras and Lie groups (that post was focused over the reals instead of the complexes, but much of the discussion carries over to the complex case).
A particular example of a Lie algebra is the general linear Lie algebra of linear transformations on a finite-dimensional complex vector space (or vector space for short) , with the commutator Lie bracket ; one easily verifies that this is indeed an abstract Lie algebra. We will define a concrete Lie algebra to be a Lie algebra that is a subalgebra of for some vector space , and similarly define a representation of a Lie algebra 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 every abstract Lie algebra has a faithful representation), but we will not need or prove this fact here.
Even without Ado’s theorem, though, the structure of abstract Lie algebras is very well understood. As with objects in many other algebraic categories, a basic way to understand a Lie algebra is to factor it into two simpler algebras via a short exact sequence
thus one has an injective homomorphism from to and a surjective homomorphism from to such that the image of the former homomorphism is the kernel of the latter. (To be pedantic, a short exact sequence in a general category requires these homomorphisms to be monomorphisms and epimorphisms respectively, but in the category of Lie algebras these turn out to reduce to the more familiar concepts of injectivity and surjectivity respectively.) Given such a sequence, one can (non-uniquely) identify with the vector space equipped with a Lie bracket of the form
for some bilinear maps and that obey some Jacobi-type identities which we will not record here. Understanding exactly what maps are possible here (up to coordinate change) can be a difficult task (and is one of the key objectives of Lie algebra cohomology), but in principle at least, the problem of understanding can be reduced to that of understanding that of its factors . To emphasise this, I will (perhaps idiosyncratically) express the existence of a short exact sequence (3) by the ATLAS-type notation
although one should caution that for given and , there can be multiple non-isomorphic that can form a short exact sequence with , so that is not a uniquely defined combination of and ; one could emphasise this by writing instead of , though we will not do so here. We will refer to as an extension of by , and read the notation (5) as “ is -by-“; confusingly, these two notations reverse the subject and object of “by”, but unfortunately both notations are well entrenched in the literature. We caution that the operation is not commutative, and it is only partly associative: every Lie algebra of the form is also of the form , but the converse is not true (see this previous blog post for some related discussion). As we are working in the infinitesimal world of Lie algebras (which have an additive group operation) rather than Lie groups (in which the group operation is usually written multiplicatively), it may help to think of as a (twisted) “sum” of and rather than a “product”; for instance, we have and , and also .
Special examples of extensions of by include the direct sum (or direct product) (also denoted ), which is given by the construction (4) with and both vanishing, and the split extension (or semidirect product) (also denoted ), which is given by the construction (4) with vanishing and the bilinear map taking the form
for some representation of in the concrete Lie algebra of derivations of , that is to say the algebra of linear maps that obey the Leibniz rule
for all . (The derivation algebra of a Lie algebra is analogous to the automorphism group of a Lie group , with the two concepts being intertwined by the tangent space functor from Lie groups to Lie algebras (i.e. the derivation algebra is the infinitesimal version of the automorphism group). Of course, this functor also intertwines the Lie algebra and Lie group versions of most of the other concepts discussed here, such as extensions, semidirect products, etc.)
There are two general ways to factor a Lie algebra as an extension of a smaller Lie algebra by another smaller Lie algebra . One is to locate a Lie algebra ideal (or ideal for short) in , thus , where denotes the Lie algebra generated by , and then take to be the quotient space in the usual manner; one can check that , are also Lie algebras and that we do indeed have a short exact sequence
Conversely, whenever one has a factorisation , one can identify with an ideal in , and with the quotient of by .
The other general way to obtain such a factorisation is is to start with a homomorphism of into another Lie algebra , take to be the image of , and to be the kernel . Again, it is easy to see that this does indeed create a short exact sequence:
Conversely, whenever one has a factorisation , one can identify with the image of under some homomorphism, and with the kernel of that homomorphism. Note that if a representation is faithful (i.e. injective), then the kernel is trivial and is isomorphic to .
Now we consider some examples of factoring some class of Lie algebras into simpler Lie algebras. The easiest examples of Lie algebras to understand are the abelian Lie algebras , in which the Lie bracket identically vanishes. Every one-dimensional Lie algebra is automatically abelian, and thus isomorphic to the scalar algebra . Conversely, by using an arbitrary linear basis of , we see that an abelian Lie algebra is isomorphic to the direct sum of one-dimensional algebras. Thus, a Lie algebra is abelian if and only if it is isomorphic to the direct sum of finitely many copies of .
Now consider a Lie algebra that is not necessarily abelian. We then form the derived algebra ; this algebra is trivial if and only if is abelian. It is easy to see that is an ideal whenever are ideals, so in particular the derived algebra is an ideal and we thus have the short exact sequence
The algebra is the maximal abelian quotient of , and is known as the abelianisation of . If it is trivial, we call the Lie algebra perfect. If instead it is non-trivial, then the derived algebra has strictly smaller dimension than . From this, it is natural to associate two series to any Lie algebra , the lower central series
and the derived series
By induction we see that these are both decreasing series of ideals of , with the derived series being slightly smaller ( for all ). We say that a Lie algebra is nilpotent if its lower central series is eventually trivial, and solvable if its derived series eventually becomes trivial. Thus, abelian Lie algebras are nilpotent, and nilpotent Lie algebras are solvable, but the converses are not necessarily true. For instance, in the general linear group , which can be identified with the Lie algebra of complex matrices, the subalgebra of strictly upper triangular matrices is nilpotent (but not abelian for ), while the subalgebra of upper triangular matrices is solvable (but not nilpotent for ). It is also clear that any subalgebra of a nilpotent algebra is nilpotent, and similarly for solvable or abelian algebras.
From the above discussion we see that a Lie algebra is solvable if and only if it can be represented by a tower of abelian extensions, thus
for some abelian . Similarly, a Lie algebra is nilpotent if it is expressible as a tower of central extensions (so that in all the extensions in the above factorisation, is central in , where we say that is central in if ). We also see that an extension is solvable if and only of both factors are solvable. Splitting abelian algebras into cyclic (i.e. one-dimensional) ones, we thus see that a finite-dimensional Lie algebra is solvable if and only if it is polycylic, i.e. it can be represented by a tower of cyclic extensions.
For our next fundamental example of using short exact sequences to split a general Lie algebra into simpler objects, we observe that every abstract Lie algebra has an adjoint representation , where for each , is the linear map ; one easily verifies that this is indeed a representation (indeed, (2) is equivalent to the assertion that for all ). The kernel of this representation is the center , which the maximal central subalgebra of . We thus have the short exact sequence
For our next fundamental decomposition of Lie algebras, we need some more definitions. A Lie algebra is simple if it is non-abelian and has no ideals other than and ; thus simple Lie algebras cannot be factored into strictly smaller algebras . In particular, simple Lie algebras are automatically perfect and centerless. We have the following fundamental theorem:
- (i) does not contain any non-trivial solvable ideal.
- (ii) does not contain any non-trivial abelian ideal.
- (iii) The Killing form , defined as the bilinear form , is non-degenerate on .
- (iv) is isomorphic to the direct sum of finitely many non-abelian simple Lie algebras.
We review the proof of this theorem later in these notes. A Lie algebra obeying any (and hence all) of the properties (i)-(iv) is known as a semisimple Lie algebra. The statement (iv) is usually taken as the definition of semisimplicity; the equivalence of (iv) and (i) is a special case of Weyl’s complete reducibility theorem (see Theorem 32), and the equivalence of (iv) and (iii) is known as the Cartan semisimplicity criterion. (The equivalence of (i) and (ii) is easy.)
If and are solvable ideals of a Lie algebra , then it is not difficult to see that the vector sum is also a solvable ideal (because on quotienting by we see that the derived series of must eventually fall inside , and thence must eventually become trivial by the solvability of ). As our Lie algebras are finite dimensional, we conclude that has a unique maximal solvable ideal, known as the radical of . The quotient is then a Lie algebra with trivial radical, and is thus semisimple by the above theorem, giving the Levi decomposition
expressing an arbitrary Lie algebra as an extension of a semisimple Lie algebra by a solvable algebra (and it is not hard to see that this is the only possible such extension up to isomorphism). Indeed, a deep theorem of Levi allows one to upgrade this decomposition to a split extension
although we will not need or prove this result here.
In view of the above decompositions, we see that we can factor any Lie algebra (using a suitable combination of direct sums and extensions) into a finite number of simple Lie algebras and the scalar algebra . In principle, this means that one can understand an arbitrary Lie algebra once one understands all the simple Lie algebras (which, being defined over , are somewhat confusingly referred to as simple complex Lie algebras in the literature). Amazingly, this latter class of algebras are completely classified:
- for some .
- for some .
- for some .
- for some .
- , or .
(The precise definition of the classical Lie algebras and the exceptional Lie algebras will be recalled later.)
(One can extend the families of classical Lie algebras a little bit to smaller values of , but the resulting algebras are either isomorphic to other algebras on this list, or cease to be simple; see this previous post for further discussion.)
This classification is a basic starting point for the classification of many other related objects, including Lie algebras and Lie groups over more general fields (e.g. the reals ), as well as finite simple groups. Being so fundamental to the subject, this classification is covered in almost every basic textbook in Lie algebras, and I myself learned it many years ago in an honours undergraduate course back in Australia. The proof is rather lengthy, though, and I have always had difficulty keeping it straight in my head. So I have decided to write some notes on the classification in this blog post, aiming to be self-contained (though moving rapidly). There is no new material in this post, though; it is all drawn from standard reference texts (I relied particularly on Fulton and Harris’s text, which I highly recommend). In fact it seems remarkably hard to deviate from the standard routes given in the literature to the classification; I would be interested in knowing about other ways to reach the classification (or substeps in that classification) that are genuinely different from the orthodox route.