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 ${\mathcal{A}}$ is the Lie algebra with generators ${\left\{a_j, j \in \mathbb{Z}\right\}}$ and another generator ${K}$. The commutation relations are

$\displaystyle [a_j, a_k] = \begin{cases} 0 & \text{ if } j + k \neq 0 \\ j K & \text{ if } j = -k . \end{cases} ,$

and we require ${K}$ to be central. This is a graded Lie algebra with ${a_j}$ in degree ${j}$ and ${K}$ 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 ${\mathbb{C} K}$.

The factor of ${j}$ in the relation for ${[a_j, a_k]}$ is of course a moot point, as we could choose a different basis so that the relation read ${[a_j, a_k] = \delta_{j, -k}}$. (The exception is ${a_0}$: that has to stay central.) However, there is a geometric interpretation of ${\mathcal{A}}$ with the current normalization. We have

$\displaystyle [a_j, a_k] = \mathrm{Res}( t^j d t^k)_{t = 0} K.$

Here ${\mathrm{Res}}$ denotes the residue of the differential form ${t^j dt^k = k t^j t^{k-1} dt}$ at ${t = 0}$. Since only terms of the form ${t^{-1} dt}$ 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:

$\displaystyle \mathcal{A}= \mathbb{C}[t, t^{-1}] \oplus \mathbb{C}K$

where ${K}$ is central, and where the Lie bracket of Laurent polynomials ${f, g }$ is

$\displaystyle \mathrm{Res}_{t =0 } (f dg) K.$

Note that any exact form has residue zero, so ${\mathrm{Res}_{t = 0}(fdg) = -\mathrm{Res}_{t=0}(g df)}$ (by comparing with ${d(gf)}$). 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 ${\mathbb{C}[t, t^{-1}]}$.

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 ${\mathbb{C}[t, t^{-1}]}$. There is another fruitful way of looking at it, which produces representations. The commutator relations

$\displaystyle [a_i, a_{-i}] = i K_0$

are (up to a constant) the canonical commutation relations between ${ \frac{\partial}{\partial x}}$ and multiplication by ${x}$, at least if we treat ${K_0}$ as the identity. The various ${\left\{a_i\right\}}$ 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 ${\mathcal{A}}$.

Definition 1 The Fock representation ${F_\mu}$ of ${\mathcal{A}}$ is the infinite polynomial ring ${\mathbb{C}[x_1, \dots, ]}$.The variables ${a_i, i \geq 1}$ act by the operators ${ i \frac{\partial}{\partial x_i}}$. The variables ${a_{-i}, i \geq 1}$ act by multiplication by ${x_i}$. The variable ${K}$ acts as the identity, and ${a_0}$ acts by ${\mu}$.

By the standard commutation relations between differentiation and multiplication, this is in fact a representation of ${\mathcal{A}}$.

There are several special properties of ${F_\mu}$. First, although ${F_\mu}$ is infinite-dimensional, it is not too bad. It is a gradedrepresentation,

$\displaystyle F_\mu = \bigoplus_i (F_{\mu})_i$

such that ${x_i}$ has degree ${-i}$. Then all the pieces are finite-dimensional.

The representation ${F_\mu}$ is, moreover, a highest-weight representation of ${\mathcal{A}}$. If we take the vector ${1 \in F_\mu}$ (that is, the polynomial ${1}$), then ${1}$ is in top degree, and consequently

$\displaystyle a_i 1 = 0, \quad i >0.$

It is also easy to see that ${a_i, i \geq 1}$ acts locally nilpotently on ${F_\mu}$.

Definition 2 A vector ${v}$ in a representation ${V}$ of ${\mathcal{A}}$ is a highest weight vector if ${a_i v = 0}$, for ${i > 0}$.

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 3 The Fock representation ${F_\mu}$ is a Verma module for ${\mathcal{A}}$: that is, if ${V}$ is any representation of ${\mathcal{A}}$ and ${v \in V}$ a vector such that ${a_i v = 0, i > 0}$, ${Kv = v, a_0 v = \mu v}$, then there is a unique homomorphism

$\displaystyle F_\mu \rightarrow V$

sending ${1 \mapsto v}$.

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

$\displaystyle a_{-1}^{n_1} a_{-2}^{n_2} \dots$

(which go to a finite length), times the highest weight vector. But ${F_\mu}$ has this property. Thus, if we construct the “true” Verma module (i.e. the one with the specified universal property), it will map to ${F_\mu}$ because ${F_\mu}$ has the highest weight vector ${1}$, and a dimension count will show that the map is an isomorphism.

These representations have an additional property:

Proposition 4 Each ${F_\mu}$ is an irreducible representation of ${\mathcal{A}}$.

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 ${P(x_1, \dots, )}$, we can use a number of differential operators to obtain a nonzero constant from it. In this way, we find that the ${\mathcal{A}}$-submodule generated by ${P(x_1, \dots, )}$ contains the highest weight vector ${1}$, which clearly generates ${F_\mu}$ (by multiplying with ${a_{-i}, i \geq 1}$).

The theory of representations of ${\mathcal{A}}$ is quite complicated. In fact, the enveloping algebra of ${\mathcal{A}}$ with ${a_0, K}$ identified with ${1}$ 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 ${W}$ is the Lie algebra of vector fields with Laurent polynomial coefficients, i.e. those of the form

$\displaystyle p(t) \frac{d}{dt} , \quad p(t) \in \mathbb{C}[t, t^{-1}].$

Consequently, we can take as a basis for ${W}$ the elements ${L_n = - t^{n+1} \frac{d}{dt}}$ (for ${n \in \mathbb{Z}}$) and with commutation relations

$\displaystyle [L_m, L_n] = (m-n) L_{m+n}.$

The Witt algebra acts on the Lie algebra ${\mathcal{A}}$; that is, there is a homomorphism of Lie algebras

$\displaystyle \eta: W \rightarrow \mathrm{Der}(\mathcal{A}).$

We let ${p(t) \frac{d}{dt}}$ act on a Laurent polynomial ${q(t) \in \mathbb{C}[t, t^{-1}]}$ via Lie derivative

$\displaystyle \eta( p(t) \frac{d}{dt})( q(t)) = p(t) q'(t).$

It acts trivially on ${K \in \mathcal{A}}$.

To motivate this construction, let’s do a slightly more complicated one. The group ${\mathrm{Diff}^+(S^1)}$ of orientation-preserving diffeomorphisms of the circle is an infinite-dimensional Lie group; its Lie algebra is the Lie algebra of vector fields on ${S^1}$.

We can construct a vector space ${\mathrm{Fun}(S^1) }$ such that elements are functions ${f: S^1 \rightarrow \mathbb{C}}$. Define a pairing

$\displaystyle ( \cdot, \cdot): \mathrm{Fun}(S^1) \times \mathrm{Fun}(S^1) \rightarrow \mathbb{C}$

sending

$\displaystyle (f, g) = \frac{1}{2 \pi i } \int_{S^1} f dg.$

This is a skew-symmetric map, because ${f dg + gdf}$ is exact (i.e. ${d(fg)}$) on ${S^1}$ and consequently has integral zero. As a result, we can define a Lie algebra

$\displaystyle \hat{\mathcal{A}} = \mathrm{Fun}(S^1, \mathbb{C}) \oplus \mathbb{C}K$

which should be thought of as a completion of the Heisenberg algebra ${\mathcal{A}}$, in some sense. The element ${K}$ is central, and the bracket of functions ${f,g}$ is given by ${(f, g)}$.

Proposition 5 The group ${\mathrm{Diff}^+(S^1)}$ of diffeomorphisms acts on the Lie algebra ${\hat{\mathcal{A}}}$.

In fact, this is now easy, since orientation-preserving diffeomorphisms of the circle preserve the integral. So if ${f, g}$ are functions on ${S^1}$ and ${\phi: S^1 \rightarrow S^1}$ is an orientation-preserving diffeomorphism, we have

$\displaystyle (f,g) = \frac{1}{2\pi i } \int_{S^1}f dg = \frac{1}{2 \pi i } \int_{S^1} \phi^* (f dg) = \frac{1}{2 \pi i } \int_{S^1} f \circ \phi d(g \circ \phi).$

As a result, the action of ${\mathrm{Diff}^+(S^1)}$ is defined on functions by precomposing; it does nothing to ${K}$.

Consequently, we get an action of the Lie algebra of vector fields on ${S^1}$ on ${\hat{\mathcal{A}}}$ such that ${K}$ 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 ${W}$ in the complexification of the Lie algebra of vector fields, and consequently ${W}$ acts on ${\hat{\mathcal{A}}}$. The restriction of this action to ${\mathcal{A}}$ is precisely the homomorphism ${\eta: W \rightarrow \mathrm{Der}(\mathcal{A})}$ defined above.

This discussion in particular shows that ${\eta}$ is in fact a homomorphism of Lie algebras, and each ${\eta(p(t) \frac{d}{dt})}$ 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 $F_\mu$. 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 $F_\mu$. This is the motivation for the Virasoro algebra.