| Title:
|
A $\mathbb Z_4^3$-grading on a $56$-dimensional simple structurable algebra and related fine gradings on the simple Lie algebras of type $E$ (English) |
| Author:
|
Aranda-Orna, Diego |
| Author:
|
Elduque, Alberto |
| Author:
|
Kochetov, Mikhail |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
55 |
| Issue:
|
3 |
| Year:
|
2014 |
| Pages:
|
285-313 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We describe two constructions of a certain $\mathbb Z_4^3$-grading on the so-called Brown algebra (a simple structurable algebra of dimension $56$ and skew-dimension $1$) over an algebraically closed field of characteristic different from $2$. The Weyl group of this grading is computed. We also show how this grading gives rise to several interesting fine gradings on exceptional simple Lie algebras of types $E_6$, $E_7$ and $E_8$. (English) |
| Keyword:
|
graded algebra |
| Keyword:
|
structurable algebra |
| Keyword:
|
exceptional simple Lie algebra |
| MSC:
|
17A30 |
| MSC:
|
17B25 |
| MSC:
|
17B70 |
| MSC:
|
17C40 |
| idZBL:
|
Zbl 06391544 |
| idMR:
|
MR3225611 |
| . |
| Date available:
|
2015-01-19T10:48:11Z |
| Last updated:
|
2016-10-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143809 |
| . |
| Reference:
|
[AM99] Albuquerque H., Majid S.: Quasialgebra structure of the octonions.J. Algebra 220 (1999), no. 1, 188–224. Zbl 0999.17006, MR 1713433, 10.1006/jabr.1998.7850 |
| Reference:
|
[All78] Allison B.N.: A class of nonassociative algebras with involution containing the class of Jordan algebras.Math. Ann. 237 (1978), 133–156. Zbl 0368.17001, MR 0507909, 10.1007/BF01351677 |
| Reference:
|
[All79] Allison B.N.: Models of isotropic simple Lie algebras.Comm. Algebra 7 (1979), no. 17, 1835–1875. Zbl 0422.17006, MR 0547712, 10.1080/00927877908822432 |
| Reference:
|
[All90] Allison B.N.: Simple structurable algebras of skew-dimension one.Comm. Algebra 18 (1990), no. 4, 1245–1279. Zbl 0702.17002, MR 1059949, 10.1080/00927879008823963 |
| Reference:
|
[AF84] Allison B.N., Faulkner J.R.: A Cayley–Dickson process for a class of structurable algebras.Trans. Amer. Math. Soc. 283 (1984), no. 1, 185–210. Zbl 0543.17015, MR 0735416, 10.1090/S0002-9947-1984-0735416-2 |
| Reference:
|
[AF92] Allison B.N., Faulkner J.R.: Norms on structurable algebras.Comm. Algebra 20 (1992), no. 1, 155–188. Zbl 0753.17004, MR 1145331, 10.1080/00927879208824337 |
| Reference:
|
[AF93] Allison B.N., Faulkner J.R.: Nonassociative coefficient algebras for Steinberg unitary Lie algebras.J. Algebra 161 (1993), no. 1, 1–19. Zbl 0812.17002, MR 1245840, 10.1006/jabr.1993.1202 |
| Reference:
|
[Bro63] Brown R.B.: A new type of nonassociative algebra.Proc. Nat. Acad. Sci. U.S.A. 50 (1963), 947–949. MR 0158913, 10.1073/pnas.50.5.947 |
| Reference:
|
[DE13] Draper C., Elduque A.: Maximal finite abelian subgroups of $E_8$.arXiv:1312.4326. |
| Reference:
|
[DV12] Draper C., Viruel A.: Fine gradings on $\mathfrak{e}_6$.arXiv:1207.6690. |
| Reference:
|
[Eld13] Elduque A.: Fine gradings and gradings by root systems on simple Lie algebras.to appear in Rev. Mat. Iberoam.; arXiv:1303.0651. |
| Reference:
|
[EK13] Elduque A., Kochetov M.: Gradings on simple Lie algebras.Mathematical Surveys and Monographs 189, American Mathematical Society, Providence, RI, 2013. Zbl 1281.17001, MR 3087174 |
| Reference:
|
[EO07] Elduque A., Okubo S.: Lie algebras with $S_4$-action and structurable algebras.J. Algebra 307 (2007), no. 2, 864–890. Zbl 1172.17013, MR 2275376, 10.1016/j.jalgebra.2006.08.033 |
| Reference:
|
[Gar01] Garibaldi S.: Structurable Algebras and Groups of Type $E_6$ and $E_7$.J. Algebra 236 (2001), no. 2, 651–691. MR 1813495, 10.1006/jabr.2000.8514 |
| Reference:
|
[Jac68] Jacobson N.: Structure and representations of Jordan algebras.American Mathematical Society Colloquium Publications 39, American Mathematical Society, Providence, RI, 1968. Zbl 0218.17010, MR 0251099 |
| Reference:
|
[McC69] McCrimmon K.: The Freudenthal-Springer-Tits constructions of exceptional Jordan algebras.Trans. Amer. Math. Soc. 139 (1969), 495–510. Zbl 0175.02703, MR 0238916, 10.1090/S0002-9947-1969-0238916-9 |
| Reference:
|
[McC70] McCrimmon K.: The Freudenthal-Springer-Tits constructions revisited.Trans. Amer. Math. Soc. 148 (1970), 293–314. Zbl 0224.17013, MR 0271181, 10.1090/S0002-9947-1970-0271181-3 |
| Reference:
|
[Smi92] Smirnov O.N.: Simple and semisimple structurable algebras.Proceedings of the International Conference on Algebra, Part 2 (Novosibirsk, 1989), Contemp. Math. 131, Part 2, Amer. Math. Soc., Providence, RI, 1992, pp. 685–694. Zbl 0765.17003, MR 1175866 |
| . |
