In the classification of Coxeter groups, or equivalently root systems:
$$A_n, B_n/C_n, D_n, E_6, E_7, E_8, F_4, G_2, H_2, H_3, H_4, I_2(p)$$
with $p \geq 7$, the last four fail to generate any simple finite dimensional Lie algebras over fields of characteristic zero because of the crystallographic restriction theorem.
However, I know that there exist well defined “projections” (foldings in the context of Coxeter-Dynkin diagrams) $A_4 \rightarrow H_2, D_6 \rightarrow H_3, E_8 \rightarrow H_4$ and $A_n \rightarrow I_2(n+1)$, and that they have been used in certain situations to extend some of the objects related to crystallographic root systems to noncrystallographic ones, or to better understand them (e.g., quasicrystals, icosians, aperiodic tilings, affine Toda field theories, etc.). This leads me to wonder whether one could define a generalization of Lie group/Lie algebra such as to include the remaining four groups in the classification. So my question is:
Is there an “almost-Lie algebra” mathematical structure corresponding to the $H_n$ and $I_2(p)$ noncrystallographic root systems, in the same way that simple Lie algebras correspond to crystallographic ones? Has this been explored somewhere?
There is a program around this called Spetses. The aim is to find some Lie theoretic object whose `Weyl group’ is a complex reflection group (which includes the non-crystallographic reflection groups above). I am not entirely clear on what has been accomplished. The notes linked allude to some results for general Coxeter groups, but I haven’t looked into those results.