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. (more…)