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. (more…)