TITLE: Does there exist something like an $H_3$ and $H_4$ (icosahedral) Lie algebra or algebraic group?
QUESTION [18 upvotes]: 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).

REPLY [6 votes]: 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.