| Title:
|
Gosset polytopes in integral octonions (English) |
| Author:
|
Chang, Woo-Nyoung |
| Author:
|
Lee, Jae-Hyouk |
| Author:
|
Lee, Sung Hwan |
| Author:
|
Lee, Young Jun |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
64 |
| Issue:
|
3 |
| Year:
|
2014 |
| Pages:
|
683-702 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study the integral quaternions and the integral octonions along the combinatorics of the $24$-cell, a uniform polytope with the symmetry $D_{4}$, and the Gosset polytope $4_{21}$ with the symmetry $E_{8}$. We identify the set of the unit integral octonions or quaternions as a Gosset polytope $4_{21}$ or a $24$-cell and describe the subsets of integral numbers having small length as certain combinations of unit integral numbers according to the $E_{8}$ or $D_{4}$ actions on the $4_{21}$ or the $24$-cell, respectively. Moreover, we show that each level set in the unit integral numbers forms a uniform polytope, and we explain the dualities between them. In particular, the set of the pure unit integral octonions is identified as a uniform polytope $2_{31}$ with the symmetry $E_{7}$, and it is a dual polytope to a Gosset polytope $3_{21}$ with the symmetry $E_{7}$ which is the set of the unit integral octonions with $\operatorname {Re}=1/2$. (English) |
| Keyword:
|
integral octonion |
| Keyword:
|
24-cell |
| Keyword:
|
Gosset polytope |
| MSC:
|
06B99 |
| MSC:
|
11Z05 |
| MSC:
|
52B20 |
| idZBL:
|
Zbl 06391519 |
| idMR:
|
MR3298554 |
| DOI:
|
10.1007/s10587-014-0126-5 |
| . |
| Date available:
|
2014-12-19T16:03:06Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144052 |
| . |
| Reference:
|
[1] Baez, J. C.: The octonions.Bull. Am. Math. Soc., New. Ser. 39 (2002), 145-205 errata ibid. 42 (2005), 213. MR 1886087, 10.1090/S0273-0979-01-00934-X |
| Reference:
|
[2] Conway, J. H., Smith, D. A.: On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry.A K Peters Natick (2003). Zbl 1098.17001, MR 1957212 |
| Reference:
|
[3] Coxeter, H. S. M.: Integral Cayley numbers.Duke Math. J. 13 (1946), 561-578. Zbl 0063.01004, MR 0019111, 10.1215/S0012-7094-46-01347-6 |
| Reference:
|
[4] Coxeter, H. S. M.: Regular and semi-regular polytopes II.Math. Z. 188 (1985), 559-591. Zbl 0547.52005, MR 0774558, 10.1007/BF01161657 |
| Reference:
|
[5] Coxeter, H. S. M.: Regular and semi-regular polytopes III.Math. Z. 200 (1988), 3-45. Zbl 0633.52006, MR 0972395 |
| Reference:
|
[6] Coxeter, H. S. M.: Regular Complex Polytopes. 2nd ed.Cambridge University Press New York (1991). Zbl 0732.51002, MR 1119304 |
| Reference:
|
[7] Coxeter, H. S. M.: The evolution of Coxeter-Dynkin diagrams.Nieuw Arch. Wiskd. 9 (1991), 233-248. Zbl 0759.20013, MR 1166143 |
| Reference:
|
[8] Koca, M.: $E_8$ lattice with icosians and $Z_5$ symmetry.J. Phys. A, Math. Gen. 22 (1989), 4125-4134. MR 1016766, 10.1088/0305-4470/22/19/007 |
| Reference:
|
[9] Koca, M., Koç, R.: Automorphism groups of pure integral octonions.J. Phys. A, Math. Gen. 27 (1994), 2429-2442. Zbl 0835.22006, MR 1279958, 10.1088/0305-4470/27/7/021 |
| Reference:
|
[10] Koca, M., Ozdes, N.: Division algebras with integral elements.J. Phys. A, Math. Gen. 22 (1989), 1469-1493. Zbl 0686.22005, MR 1003076, 10.1088/0305-4470/22/10/006 |
| Reference:
|
[11] Lee, J.-H.: Configurations of lines in del Pezzo surfaces with Gosset polytopes.Trans. Amer. Math. Soc. 366 (2014), 4939-4967. MR 3217705, 10.1090/S0002-9947-2014-06098-4 |
| Reference:
|
[12] Lee, J.-H.: Gosset polytopes in Picard groups of del Pezzo surfaces.Can. J. Math. 64 (2012), 123-150. Zbl 1268.14038, MR 2932172, 10.4153/CJM-2011-063-6 |
| . |
