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.

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