Does there exist something like an $H_3$ and $H_4$ (icosahedral) Lie algebra or algebraic group?

Question

The (finite-dimensional) complex simple Lie algebras have been classified by Killing and Cartan a long time ago in the $\mathsf{A}_n,\mathsf{B}_n,\mathsf{C}_n,\mathsf{D}_n$ families and $\mathsf{G}_2,\mathsf{F}_4,\mathsf{E}_6,\mathsf{E}_7,\mathsf{E}_8$ exceptional cases. So the non-crystallographic $\mathsf{H}_3$ and $\mathsf{H}_4$ Coxeter groups (i.e., icosahedral symmetry) do not appear as Weyl groups of such algebras.

On the other hand, there is also no such thing as a field with one element, yet it is interesting to construct something which might play that role. Perhaps more relevantly here, there is no such thing as a tiling of the plane with regular pentagons, but there is something of the sort. And the concept of finite-dimensional simple Lie algebra can be generalized in various ways, e.g., Kac-Moody algebras.

So I've often wanted to ask: is there something which is remotely like the $\mathsf{H}_3$ and $\mathsf{H}_4$ Lie algebras or algebraic groups? They should probably be $33$- and $124$-dimensional respectively, and certainly the golden ratio $\frac{1+\sqrt{5}}{2}$ should play an important role in their definition (posing as an integer).

Answer 1

One potential answer to your question is given by the theory of $p$-compact groups. See

Grodal, The classification of p-compact groups and homotopical group theory. Proc. Intl. Congress of Mathematicians 2010 (Hyderabad, 2010), Volume II, 973–1001.

and the references therein. The notion of a p-compact group is homotopical, and there is a classification in terms of root data over $\mathbb{Z}_p$ (once it has been defined what this means).

In Table 1 of the above paper one can can see a classification of the irreducible $\mathbb{Q}_p$ reflection groups. In particular the dihedral groups (line 2), as well as $H_3$ ("$G_{23}$" in the table) and $H_4$ ("$G_{30}$") all occur, with the later two definable over $\mathbb{Q}_p$ if and only if $p = 1$ or $4$ modulo 5.

Answer 2

As far as I know, nobody has proposed candidates for such Lie algebras or algebraic groups. From the viewpoint of Chevalley's old work on structure and classification (over an arbitrary algebraically closed field), it's hard to see what kind of algebraic group would fit.

On the other hand, it's useful to be aware of the natural embeddings of the non-crystallographic finite (real) reflection groups into crystallographic ones (Weyl groups). See for example the notes on page 48 of my 1990 textbook on reflection groups, expanded somewhat in my list of revisions here. The earliest source I know is a 1979 paper (MR) by Sekiguchi and Yano showing how to embed $H_3$ into the Weyl group $D_6$ via a natural "folding" process of Coxeter graphs. In 3.9(b) of a 1983 paper freely available here, Lusztig similarly showed how to embed $H_4$ into $E_8$, justified by an application of his Hecke algebra formalism. A posthumous 1988 paper by Oleg Shcherbak here gives the most comprehensive and elementary account of such embeddings, including even the non-crystallographic dihedral types. (He was a student of V. Arnold in Moscow.)

Note that the folding process here is a somewhat more devious version of the popular foldings in Lie theory which correlate well with fixed point subgroups: for example, $E_6$ folds to $F_4$. In all these foldings, the Coxeter numbers of the finite reflection groups involved coincide. But I don't know any plausible way to extract from the embeddings of non-crystallographic types any associated finite dimensional Lie algebras (or algebraic groups). Maybe look at Kac-Moody theory or such?

Answer 3

I will address an "easier question", namely analogues of algebraic groups corresponding to finite dihedral groups $I_2(n)$ of non-crystallographic type. At least in this case, we have candidate groups, but their theory was never developed much.

First of all, Tits had realized in 1950s that semisimple algebraic groups (more precisely, sets of $F$-points of such groups for fields $F$) can be realized as automorphism groups of certain simplicial complexes, called (thick) spherical buildings modeled on Coxeter complexes of finite Coxeter groups $W$. In 1977 Tits proved existence of such buildings for all finite dihedral groups $I_2(n)$. It is known (it might be already in Tits' 1977 work, I forgot) that buildings for $I_2(n)$ constructed by Tits (provided that some care is taken during the construction) have large groups of automorphisms $G$, namely, they are flag-transitive. At the very least, this follows from the work of Tent. In particular, these groups $G$ have structure of BN pairs, etc. However, algebraic nature (in the sense of connection to algebraic geometry and Lie algebras) of groups $G$ remains a mystery. Some circumstantial evidence that such connection exists appears in my paper with Arkady Berenstein (where you can also find references to the relevant work of Tits and of Tent):

A. Berenstein, M. Kapovich, Stability inequalities and universal Schubert calculus of rank 2, Transformation Groups, Vol. 16 (2011) p. 955-1007.

Now, the bad news is that Tits also proved that there are no thick spherical buildings for the Coxeter groups $H_3$ and $H_4$. It is quite possible that one needs to relax spherical building axioms but at this stage it is very much unclear how in order to make a meaningful connection to the group theory.