# Heisenberg subalgebras of affine Lie algebras

It seems to be "well-known" that (infinite-dimensional) Heisenberg subalgebras of an affine Lie algebra $\hat{\mathfrak{g}}$ corresponding to a finite-dimensional simple Lie algebra $\mathfrak{g}$ of type $A$, $D$, or $E$ are parametrized (up to conjugacy) by the set of conjugacy classes in the Weyl group of $\mathfrak{g}$. For instance, this statement can be found in the paper "112 constructions of the basic representation of the loop group of $E_8$" by Kac and Peterson (MR). However, no proof is given in this paper. (A Heisenberg subalgebra is constructed for each element of the Weyl group, but the assertion that this leads to a parametrization of all of them is not proved.) Does anyone know of a reference (with proof) of this classification of Heisenberg subalgebras of affine Lie algebras?

• It would be useful to give a definition of "Heisenberg subalgebras". Do you mean 3-dimensional? There are several generalizations, the most common being the $(2n+1)$-dimensional nilpotent Lie algebra with derived subalgebra of dimension 1. – YCor Feb 13 '14 at 15:49
• I mean the infinite-dimensional Heisenberg algebra. I've edited the post. – Alistair Savage Feb 13 '14 at 16:21
• OK, well, I can think of one way to define an infinite-dim Heisenberg Lie algebra but I can't be sure it's the only one since in Lie algebras, different communities tend to systematically use different terminologies. Consider the Lie algebra with basis $x_i,y_j$,z ($i,j$ in a countable index set $I$) with $[x_i,y_i]=z$, other brackets zero. Is this what you refer to? – YCor Feb 13 '14 at 16:41
• @Yves: Yes, that's the one. – Alistair Savage Feb 13 '14 at 16:44

## 2 Answers

Up to the central extension (which doesn't affect the classification) this is a special case of the classification of Cartan subgroups of a reductive group over a field. Since they all split after an etale extension, where they are uniquely conjugate up to Weyl group, they are given by Galois $H^1$ of the field with coefficients in the Weyl group. In our case the Galois group is $\widehat {\mathbf Z}$, so we get just elements of $W$ up to conjugacy. Geometrically this means such a subgroup is conjugate to standard one after passing to a branched cover of the disc (etale cover of punctured disc), and so is classified by the monodromy of the associated cameral cover ($W$-torsor over the punctured disc of all ways to conjugate to a standard Cartan).

• Thanks, David. I can't help but think that there's a simpler (i.e. more elementary) proof. People seem to always refer to the Kac-Peterson paper for this result, and that paper doesn't mention Galois cohomology, etale extensions, etc. – Alistair Savage Feb 17 '14 at 0:27
• Thanks for the nice answer David. Could you please explain a bit why taking central extensions does not affect classification? I would say that this proof is relatively elementary. Probably one can phrase the whole thing without mentioning cohomology. The proof appears, e.g., in Kazhdan and Lusztig's paper "Fixed point varieties on affine flag manifolds" which was published just a couple of years after Kac-Peterson. These ideas were presumably known long before. – Dr. Evil Dec 11 '15 at 21:31

Kac and Peterson have classified all inequivalent Heisenberg subalgebras of a loop algebra. This is explained in the book "Lie Algebras, Part 2: Finite and Infinite Dimensional Lie Algebras and Applications in Physics" by de Kerf, Bäuerle and ten Kroode, chapter $26$.

• The book you mention is quite useful as a reference (thanks!). However, it doesn't seem to contain the proof I'm looking for. Instead, it refers to the paper by Kac and Peterson that I mentioned in my post, which seems to state the result without proof. – Alistair Savage Feb 13 '14 at 16:38