# 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, 2014 at 15:49
• I mean the infinite-dimensional Heisenberg algebra. I've edited the post. Feb 13, 2014 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, 2014 at 16:41
• @Yves: Yes, that's the one. Feb 13, 2014 at 16:44

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).
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 $26$.