Root graded groups of type $H_3$ and $H_4$

Abstract

Using the well-known realisation of the root system $H_4$ as a folding of $E_8$, one can construct examples of $H_4$-graded groups from Chevalley groups of type $E_8$. Such Chevalley groups are defined over a commutative ring $R$, and the root groups of the resulting $H_4$-grading are coordinatised by $R\times R$. We show that every $H_4$-graded group arises as the folding of an $E_8$-graded group, or in other words, that it is coordinatised by $R\times R$ for some commutative ring $R$. We also prove similar assertions for $(D_6,H_3)$ in place of $(E_8,H_4)$.

v1: 42 pages, v2: 63 pages, changes: Improved the presentation, in particular in Section 4. Added more details for the case $H_4$. Added Section 7. Moved some material from old Section 8 to new Section 10 and from Section 4 to Appendix A, v3: 63 pages, changes: Added DOI and journal reference on the first page. Corrected some minor errors