Apologies for the lack of posts here lately; I’ve been meaning to say many things that I simply have not gotten around to doing. I’ve been taking a course on infinite-dimensional Lie algebras this semester. There are a number of important examples here, most of which I’ve never seen before. This post will set down two of the most basic.

**1. The Heisenberg algebra**

The **Heisenberg Lie algebra** is the Lie algebra with generators and another generator . The commutation relations are

and we require to be central. This is a *graded* Lie algebra with in degree and in degree zero.

The Heisenberg algebra is a simple example of a nilpotent Lie algebra: in fact, it has the property that its center contains the commutator subalgebra .

The factor of in the relation for is of course a moot point, as we could choose a different basis so that the relation read . (The exception is : that has to stay central.) However, there is a geometric interpretation of with the current normalization. We have

Here denotes the residue of the differential form at . Since only terms of the form contribute to the residue, this is easy to check.

As a result, we can think of as the Lie algebra of Laurent polynomials plus a one-dimensional space:

where is central, and where the Lie bracket of Laurent polynomials is

Note that any exact form has residue zero, so (by comparing with ). This explains the antisymmetry of the above form.

The Heisenberg algebra is thus a one-dimensional central extension of the abelian (and thus somewhat uninteresting) Lie algebra .

**2. Representations of the Heisenberg algebra**

We showed that the Heisenberg algebra could be viewed as a one-dimensional central extension of the abelian Lie algebra . There is another fruitful way of looking at it, which produces representations. The commutator relations

are (up to a constant) the canonical commutation relations between and multiplication by , at least if we treat as the identity. The various all commute with each other, so we can think of them as coming from different variables.

As a result, we can define a representation of .

Definition 1TheFock representationof is the infinite polynomial ring .The variables act by the operators . The variables act by multiplication by . The variable acts as the identity, and acts by .

By the standard commutation relations between differentiation and multiplication, this is in fact a representation of .

There are several special properties of . First, although is infinite-dimensional, it is not too bad. It is a *graded*representation,

such that has degree . Then all the pieces are finite-dimensional.

The representation is, moreover, a **highest-weight representation** of . If we take the vector (that is, the polynomial ), then is in top degree, and consequently

It is also easy to see that acts locally nilpotently on .

Definition 2A vector in a representation of is ahighest weight vectorif , for .

The analogy from the theory of finite-dimensional semisimple Lie algebras is evident. There, one has a theory of highest-weight modules, and the “universal” ones are the so-called Verma modules. One can formulate an analog in this case; it turns out, though, that we have already met the Verma modules.

Proposition 3The Fock representation is a Verma module for : that is, if is any representation of and a vector such that , , then there is a unique homomorphism

sending .

To see this, note that a defining property of the Verma module is that it has a basis on the ordered monomials

(which go to a finite length), times the highest weight vector. But has this property. Thus, if we construct the “true” Verma module (i.e. the one with the specified universal property), it will map to because has the highest weight vector , and a dimension count will show that the map is an isomorphism.

These representations have an additional property:

Proposition 4Each is an irreducible representation of .

In general, for a graded Lie algebra, Verma modules will not usually be irreducible, though they will be “generically.” (In the finite-dimensional semisimple Lie algebra case, for very specific parameters, Verma modules turn out to have finite-dimensional irreducible quotients, which give the finite-dimensional irreducible representations.) But in this case that happens.

The proof is straightforward: given any nonzero polynomial , we can use a number of differential operators to obtain a nonzero constant from it. In this way, we find that the -submodule generated by contains the highest weight vector , which clearly generates (by multiplying with ).

The theory of representations of is quite complicated. In fact, the enveloping algebra of with identified with is the Weyl algebra of differential operators in infinitely many variables, and this gives rise to the theory of D-modules, about which I can’t say much.

**3. The Witt algebra**

The **Witt algebra** is the Lie algebra of vector fields with Laurent polynomial coefficients, i.e. those of the form

Consequently, we can take as a basis for the elements (for ) and with commutation relations

The Witt algebra acts on the Lie algebra ; that is, there is a homomorphism of Lie algebras

We let act on a Laurent polynomial via Lie derivative

It acts trivially on .

To motivate this construction, let’s do a slightly more complicated one. The group of orientation-preserving diffeomorphisms of the circle is an infinite-dimensional Lie group; its Lie algebra is the Lie algebra of vector fields on .

We can construct a vector space such that elements are functions . Define a pairing

sending

This is a skew-symmetric map, because is exact (i.e. ) on and consequently has integral zero. As a result, we can define a Lie algebra

which should be thought of as a completion of the Heisenberg algebra , in some sense. The element is central, and the bracket of functions is given by .

Proposition 5The group of diffeomorphisms acts on the Lie algebra .

In fact, this is now easy, since orientation-preserving diffeomorphisms of the circle preserve the integral. So if are functions on and is an orientation-preserving diffeomorphism, we have

As a result, the action of is defined on functions by precomposing; it does nothing to .

Consequently, we get an action of the Lie algebra of vector fields on on such that is annihilated and a function goes to its Lie derivative (i.e. the usual action of a vector field on a function). Now we can imbed the Witt algebra in the complexification of the Lie algebra of vector fields, and consequently acts on . The restriction of this action to is precisely the homomorphism defined above.

This discussion in particular shows that is in fact a homomorphism of Lie algebras, and each defines a derivation; these can, of course, be checked directly.

Since the Witt algebra acts on the Heisenberg algebra by derivations, we might hope that the Witt algebra acts on the Fock representations . This is not quite true; it turns out that one has to make a one-dimensional central extension, and construct a new Lie algebra, to act on . This is the motivation for the Virasoro algebra.

March 2, 2012 at 1:21 pm

Really, that will be tough to follow. I understand it may take effort but consider giving (more) applications and history, even if trivial, and outlining where you plan to go.

The area of infinite-dimensional algebra feels like a melting pot of purely technical explorations, pipe dreams of theoretical physicists, pipe dreams (moonshines?) of mathematicians (I also think of Cherednik), wild combinatorics, and actually useful if esoteric-sounding structures. So it takes particular effort to present in a memorable way.

Also, if you can comment on the people you meet at Harvard, for instance on Pavel Etingof (which I understand gives the course), please do, anything, that would help hermits like me gain a sense of the real world.

Thanks alot for the post(s) in any case.

March 2, 2012 at 11:14 pm

I don’t know too much about history. So far I was just going to motivate some of the representation theory of the Virasoro algebra (to the extent that I can understand it). I think, though, that the theory of loop groups is the geometric counterpart to a lot of this, in that these infinite-dimensional Lie algebras are the Lie algebras of such groups.

As the course progresses I’ll probably have more to say. One of my hopes is to learn something about vertex algebras (as in Frenkel and Ben-Zvi’s book). For now my goal is to try to understand, at a relatively elementary level, why these constructions are natural and not completely ad hoc (which was my initial first impression).

I don’t really want to talk about people on a public website, but if you would have questions about mathematical culture (at least, as it is viewed by an inexperienced student), feel free to email me.

March 5, 2012 at 10:19 pm

Is there a Lie algebra map from A to W? I ask this because if so it may form a crossed module in the category of (inf.dim.) Lie algebras.

March 5, 2012 at 10:47 pm

Not that I can see. It seems that acts on , not the other way around. Elements of don’t tend to commute (it’s simple) while elements of almost always do.