Last update: 13 August 2012
This is a typed version of I.G. Macdonald's lecture notes on Kac-Moody Lie algebras from 1983.
Before we get down to serious business, let me begin by telling you a little in general terms about this subject, and what I propose to do (and what I do not propose to do) is this course of lectures. Basically it is an outgrowth of the theory of finite-dimensional complex simple (or semisimple) Lie algebras developed by Killing and E. Cartan nearly 100 years ago. If is a complex finite-dimensional semisimple Lie algebra, one can associate canonically with (I will go into details later) a certain matrix of integers, called the Cartan matrix of , which determines up to isomorphism. This matrix satisfies the following conditions:
|All principal minors of are positive.|
Conversely, any matrix of integers satisfying these two conditions is the Cartan matrix of some semisimple Lie algebra , and one can write down generators and relations for which involve only the integers . (SOMETHING GOES HERE) (I will write them down explicitly a little later.)
In the late 1960's, V. Kac and R. Moody more or less simultaneously and independently, had the idea of starting with a "generalized Cartan matrix" (satisfying but not ), writing down the same set of generators and relations as in Serr's theorem, and looking at the resulting Lie algebra (which is now infinite-dimensional). These are the so-called Kac-Moody Lie algebras, or Lie algebras defined by generalized Cartain matrices.
Of course, nearly always when one takes an attractive and elegant piece of mathematics, such as the classical theory of finite-dimensional semisimple Lie algebras over , and starts tinkering with it by weakening the axioms, the result is usually of no interest at all. The surprising thing in this case is that one does continue to get a coherent theory (which of course includes the classical theory as a special case) in which all the main features of the classical theory have their counterparts: root system, Weyl group, representations and characters – culminating in a generalization (due to V. Kac) of Weyl's character formula.
Moreover it has become clear during the last 10 years or so that these Kac-Moody Lie algebras impinge on many other areas of mathematics:
|Number Theory||(Modular forms)|
|Combinatorics||(Partitions, Rogers-Ramanujan identities)|
|Topology||(Loop spaces and Loop groups)|
|Linear Algebra||(Representations of Quivers)|
|Completely integrable systems|
|Mechanics and Particle Physics|
My aim in this course of lectures is to cover the basic structure and representation theory of Kac-Moody Lie algebras, but not any of the applications listed above. Also I don't propose to assume any particular knowledge of Lie algebras, so I will begin by briefly reviewing some of the basic notions. We shall work always over a field of characteristic 0 (and fairly soon will be , the field of complex numbers). A Lie algebra is then a vector space over endowed with a bilinear multiplication
for all (Jacobi identity).
By applying (1) to and using the bilinearity of the bracket we have
and conversely (1) (1) (take ).
Many notions for groups have counterparts for Lie algebras. This is hardly surprising, given the origin of the subject.
Let be a Lie algebra. If are vector subspaces (or just subsets) of , let denote the subspace of spanned by all with . Observe that (because ).
Subalgebra: a vector subspace of is a subalgebra if (so that is a Lie algebra in it's own right).
Ideal: a vector subspace of is an ideal of if (normal subgroup).
Quotient algebra: Let be an ideal in , and form the vector space quotient , whose elements are the cosets . Define , this does not depend on the choice of representations, and makes into a Lie Algebra.
Homomorphism: a homomorphism from to is a –linear map
for all .
It's kernel is an ideal in , it's image is a subalgebra of , and induces an isomorphism
If are such that , we say that commute. In particular, if any two elements of commute, ie if we say that is abelian (Ex. 1 above).
Centre of . It is an ideal in .
Derived algebra consists of all linear combinations of brackets . is an ideal in (by virtue of the Jacobi identity):
Moreover is abelian (and is the
smallest ideal with abelian quotient).
Derived series, upper and lower central series; nilpotent, solvable Lie algebras.
For each we define by . Then
is a homomorphism of Lie algebras, because for all we have
Moreover each is a derivation of :
This is another equivalent form of the Jacobi identity.
The kernel of ad is the centre of .
If is such that is nilpotent then we can form (Ex. 3 above) which is an automorphism of . The subgroup Int() of Aut() generated by these is the group of inner automorphisms of . It is a normal subgroup of Aut(), because if we have and therefore also
Let be a Lie algebra. A representation of on a –vector space is by definition a Lie algebra homomorphism . In other words, for each we have a linear transformation depending on linearity on :
An equivalent notion is that of a –module, which is a vector space on which acts linearly, i.e. we are given a bilinear mapping
To connect the two notions, define .
Usual notions of irreducibility, direct sums etc.
If is a group, a –module (or representation of ) is the same thing as a –module, where is the group algebra of over . The analogue of this for Lie algebras is the universal enveloping algebra of a Lie algebra , which may be defined as follows: for the tensor algebra of the vector space
Let be the two-sided ideal of IS THIS A T? generated by all
is functional in if is a homomorphism of Lie algebras, it induces , and
so that T() maps into and hence induces
U is the left adjoint of the functor L (Ex. 2) from associative algebras to Lie algebras (over ): for each Lie algebra and each associative algebra A, there is a canonical bijection
For if is a Lie algebra homomorphism, it is a –linear mapping such that
Extend to a homomorphism by defining
then the kernel of contains the ideal , by virtue of (1). Hence induces a homomorphism of associative algebras .
In the other direction, let be a homomorphism of associative algebras, and form the linear mapping
which is a Lie algebra homomorphism . Finally verify that the mappings are inverses of each other.
In particular, if is a representation, we have , i.e. is a –module.
Recall that is a graded algebra, but is not a graded ideal, because the generators are not homogeneous: . So is not a graded algebra; but it does carry a filtration, defined as follows: Let
Let be the canonical homomorphism, and let
The are vector subspaces of :
and since we have , ie is a strong filtered associative –algebra. Now form the associated graded algebra: define
then the multiplication in induces bilinear mappings
namely (for )
which make into a graded associated –algebra. For each we have
commutative with exact rows; is surjective, hence we have a surjective algebra homomorphism
In particular, if we have
Hence the kernel of contains the two-sides ideal of generated by all ; now is by definition the symmetric algebra of the vector space ; hence induces a surjective homomorphism
(P – B – W) is an isomorphism.
For the proof, see the standard texts (Bourbaki (Ch. I), Humphreys, Jacoborn)
Let be the canonical homomorphism.
Let be a vector subspace of which is mapped isomorphically by onto . Then is a complement of in .
commutes, hence maps isomorphically onto (because is an isomorphism, by P-B-W). Hence the result.
The canonical map is injective.
We may therefore identify with it's image in .
Let be a totally ordered –basis of . Then the elements
such that form a –basis of .
Let be the subspace of spanned by all with . Clearly maps isomorphically onto , hence by Corollary 7.2 is a complement of in . By induction on it follows that is the direct sum of the for all .
(Corollary 7.4 is also known as the P-B-W theorem).
To recapitulate from last time:– to each Lie algebra we associate ,
its universal enveloping algebra:
where generated by all
(by virtue of the P-B-W theorem) and we identify
and then observe that , so that induces as desired. Thus is a mapping
which one easily verifies to be bijective (i.e. is a left adjoint of the functor L, as I said last time).
Recall also (Corollary 7.4 of P-B-W th.) that if is an ordered –basis of , then the monomials with and (if , the product is empy and conventionally is said to be read as 1, the identity element of ) form a –basis of . This has the following consequence: if
where are subalgebras and is the direct sum of the vector spaces , then are subalgebras of and
(Take ordered bases of respectively; the monomials with form a –basis of , the monomials with form a –basis of , and the monomials with and form a –basis of .)
Let be a set, a field. We want to define the free Lie algebra Lie() on the set . There are two ways of proceeding: one involves P-B-W, the other doesn't.
(1) Form the free non-associative algebra F() on . How does one do this?
Define inductively sets by
and put (disjoint union) (the "free magma" on ).
If , say and , then so we have a multiplication in . Then is the –algebra with as basis, i.e. it consists of all finite linear combinations with and , and multiplication defined in the obvious way:
Now let be the 2-sided ideal in generated by all
where is a graded algebra and is a homogeneous ideal. It is clear that Lie() is a Lie algebra and that if
is the canonical embedding, then any mapping of the set into a Lie algebra extends uniquely to a Lie algebra homomorphism
(extend in the obvious way to and observe that the generators of in the kernel of , by definition).
In other words, we have a bijection
i.e. the functor Lie (from sets to Lie algebras) is a left adjoint of the forgetful functor (from Lie algebras to sets).
(2) Let be the free associative algebra on (, where is the 2-sided ideal generated by all ). The embedding
induces, as we have just seen, a Lie algebra homomorphism
But also we have a mapping , hence (by the universal property of ) a homomorphism of associative algebras
Check that these two homomorphisms are inverses of each other, hence that
By P-B-W, Lie() embeds in , hence in , so that the mapping above is injective. In other words, the free Lie algebra Lie() may be described as the subalgebra of generated by .
We have and , also , hence , hence (because injective); , hence (because is injective).
Finally, if is any subject of Lie(), the Lie algebra generated by subject to the relations is by definition , where is the ideal of the Lie() generated by (i.e. the intersection of all ideals of Lie() which contain ).
This is the classical theory we intend to generalise. A Lie algebra is said to be simple if its only ideals are 0 and , and if also is non-abelian (thus deliberately excluding the 1-dimensional abelian Lie algebra). Take , and dim .
An element is semisimple if ad is a semisimple (i.e., diagonalizable) linear transformation.
Let be simple, finite-dimensional. Then has nonzero subalgebras consisting of semisimple elements (toral subalgebras); they are necessarily abelian.
Let be a maximal toral subalgebra (or Cartan subalgebra) of . Certainly such exist, for dimensional reasons. Moreover (a non-trivial fact) any two such are conjugate in (i.e. transforms of each other under the group Int() of inner automorphisms). Fix once for all. is called the rank of .
Let be the vector space dual of . Introduce the killing form
This is symmetric, nondegenerate and invariant, ie
Moreover its restriction to is nondegenerate, hence defines an isomorphism and a symmetric bilinear form on .
Example Lie algebras of matrices with trace 0. Here we may take to consist of the diagonal matrices
(so that ).
Consider the adjoint representation of , restricted to : this is a representation of on . Since is abelian, all its irreducible representations are 1-dimensional, so that splits up into a direct sum of 1-dimensional –modules. explicitly, for each define
Then it turns out that ; the nonzero such that are called the roots of (relative to ). They form a finite subject of , called the root system of , and we have
Moreover each is 1-dimensional, and (hence is 0 if ). If is a root, so is .
In the case of , let be the matrix units, and for let be the th projection: . Then
which shows that the roots are , and the root space decomposition is clear. We compute the Killing form on as follows from above
(remember that ). So it is a multiple of the obvious scalar product.
It is possible to choose roots such that each root is of the form with coefficients and either (positive roots) or all (negative roots). The are called a set of simple roots or a basis of (they are also a basis of ). Choose such a basis once for all. There is then a unique highest root, for which is a maximum; and a unique lowest root, for which is a minimum.
For each , let denote the reflection in the hyperplane orthogonal to in , so that
is the coroot of . The reflections
corresponding to the simple roots generate
a finite group of isometries of , called the
Weyl group of
are integers, and the matrix is called the Cartan matrix of. It is independent of the choices of and of basis of . It satisfies the following conditions:
|All principal minors of are positive.|
In the case of we may take . The Weyl group is the symmetric group (for interchanges and and leaves the other fixed). Here the Cartan matrix is
(with rows and columns).
The Cartan matrix determines up to isomorphism.
Choose generators such that and elements such that , so that
Then the elements generate subject to the following relations (Serre):
The idea now is (roughly) the following: start with any matrix of integers satisfying (), and form the Lie algebra with the above generators and relations.
However there is one remark that should be made at this point. In the classical set up (which I have just been describing) the Cartan matrix is nonsingular, the form a basis of the Cartan subalgebra , and the simple roots . Now a generalized Cartan matrix may well be singular (and it would be foolish to exclude this possibility, because for the affine Lie algebras the Cartan matrix is singular).