Journals / Turkish Journal of Physics / 1998 / Cilt: 22 - Sayı: 5
Quaternionic Roots of E 8 Related Coxeter Graphs and Quasicrystals
- Journal
- Turkish Journal of Physics
- Pages
- 421–436
- DOI
- —
Abstract
The lattice matching of two sets of quaternionic roots of F4 leads to quaternionic roots of E8 which has a decomposition H4 + s H4 where the Coxeter graph H4 is represented by the 120 quaternionic elements of the binary icosahedral group. The 30 pure imaginary quaternions constitute the roots of H3 which has a natural extension to H3 + s H3 describing the root system of the Lie algebra D6. It is noted that there exist three lattices in 6-dimensions whose point group W(D6) admits the icosahedral symmetry H3 as a subgroup, the roots of which describe the mid-points of the edges of an icosahedron. A natural extension of the Coxeter group H2 of order 10 is the Weyl group W(A4) where H2 + s H2 constitute the root system of the Lie algebra A4. The relevance of these systems to quasicrystals are discussed.
Özet
The lattice matching of two sets of quaternionic roots of F4 leads to quaternionic roots of E8 which has a decomposition H4 + s H4 where the Coxeter graph H4 is represented by the 120 quaternionic elements of the binary icosahedral group. The 30 pure imaginary quaternions constitute the roots of H3 which has a natural extension to H3 + s H3 describing the root system of the Lie algebra D6. It is noted that there exist three lattices in 6-dimensions whose point group W(D6) admits the icosahedral symmetry H3 as a subgroup, the roots of which describe the mid-points of the edges of an icosahedron. A natural extension of the Coxeter group H2 of order 10 is the Weyl group W(A4) where H2 + s H2 constitute the root system of the Lie algebra A4. The relevance of these systems to quasicrystals are discussed.