A ℤ 4 3 -grading on a 56 -dimensional simple structurable algebra and related fine gradings on the simple Lie algebras of type E

Diego Aranda-Orna; Alberto Elduque; Mikhail Kochetov

Commentationes Mathematicae Universitatis Carolinae (2014)

  • Volume: 55, Issue: 3, page 285-313
  • ISSN: 0010-2628

Abstract

top
We describe two constructions of a certain 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 .

How to cite

top

Aranda-Orna, Diego, Elduque, Alberto, and Kochetov, Mikhail. "A $\mathbb {Z}_4^3$-grading on a $56$-dimensional simple structurable algebra and related fine gradings on the simple Lie algebras of type $E$." Commentationes Mathematicae Universitatis Carolinae 55.3 (2014): 285-313. <http://eudml.org/doc/261871>.

@article{Aranda2014,
abstract = {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 $E_8$.},
author = {Aranda-Orna, Diego, Elduque, Alberto, Kochetov, Mikhail},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {graded algebra; structurable algebra; exceptional simple Lie algebra; graded algebra; structurable algebra; exceptional simple Lie algebra},
language = {eng},
number = {3},
pages = {285-313},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
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 $E$},
url = {http://eudml.org/doc/261871},
volume = {55},
year = {2014},
}

TY - JOUR
AU - Aranda-Orna, Diego
AU - Elduque, Alberto
AU - Kochetov, Mikhail
TI - A $\mathbb {Z}_4^3$-grading on a $56$-dimensional simple structurable algebra and related fine gradings on the simple Lie algebras of type $E$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2014
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 55
IS - 3
SP - 285
EP - 313
AB - 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 $E_8$.
LA - eng
KW - graded algebra; structurable algebra; exceptional simple Lie algebra; graded algebra; structurable algebra; exceptional simple Lie algebra
UR - http://eudml.org/doc/261871
ER -

References

top
  1. Albuquerque H., Majid S., 10.1006/jabr.1998.7850, J. Algebra 220 (1999), no. 1, 188–224. Zbl0999.17006MR1713433DOI10.1006/jabr.1998.7850
  2. Allison B.N., 10.1007/BF01351677, Math. Ann. 237 (1978), 133–156. Zbl0368.17001MR0507909DOI10.1007/BF01351677
  3. Allison B.N., 10.1080/00927877908822432, Comm. Algebra 7 (1979), no. 17, 1835–1875. Zbl0422.17006MR0547712DOI10.1080/00927877908822432
  4. Allison B.N., 10.1080/00927879008823963, Comm. Algebra 18 (1990), no. 4, 1245–1279. Zbl0702.17002MR1059949DOI10.1080/00927879008823963
  5. Allison B.N., Faulkner J.R., 10.1090/S0002-9947-1984-0735416-2, Trans. Amer. Math. Soc. 283 (1984), no. 1, 185–210. Zbl0543.17015MR0735416DOI10.1090/S0002-9947-1984-0735416-2
  6. Allison B.N., Faulkner J.R., 10.1080/00927879208824337, Comm. Algebra 20 (1992), no. 1, 155–188. Zbl0753.17004MR1145331DOI10.1080/00927879208824337
  7. Allison B.N., Faulkner J.R., 10.1006/jabr.1993.1202, J. Algebra 161 (1993), no. 1, 1–19. Zbl0812.17002MR1245840DOI10.1006/jabr.1993.1202
  8. Brown R.B., 10.1073/pnas.50.5.947, Proc. Nat. Acad. Sci. U.S.A. 50 (1963), 947–949. MR0158913DOI10.1073/pnas.50.5.947
  9. Draper C., Elduque A., Maximal finite abelian subgroups of E 8 , arXiv:1312.4326. 
  10. Draper C., Viruel A., Fine gradings on 𝔢 6 , arXiv:1207.6690. 
  11. Elduque A., Fine gradings and gradings by root systems on simple Lie algebras, to appear in Rev. Mat. Iberoam.; arXiv:1303.0651. 
  12. Elduque A., Kochetov M., Gradings on simple Lie algebras, Mathematical Surveys and Monographs 189, American Mathematical Society, Providence, RI, 2013. Zbl1281.17001MR3087174
  13. Elduque A., Okubo S., 10.1016/j.jalgebra.2006.08.033, J. Algebra 307 (2007), no. 2, 864–890. Zbl1172.17013MR2275376DOI10.1016/j.jalgebra.2006.08.033
  14. Garibaldi S., 10.1006/jabr.2000.8514, J. Algebra 236 (2001), no. 2, 651–691. MR1813495DOI10.1006/jabr.2000.8514
  15. Jacobson N., Structure and representations of Jordan algebras, American Mathematical Society Colloquium Publications 39, American Mathematical Society, Providence, RI, 1968. Zbl0218.17010MR0251099
  16. McCrimmon K., 10.1090/S0002-9947-1969-0238916-9, Trans. Amer. Math. Soc. 139 (1969), 495–510. Zbl0175.02703MR0238916DOI10.1090/S0002-9947-1969-0238916-9
  17. McCrimmon K., 10.1090/S0002-9947-1970-0271181-3, Trans. Amer. Math. Soc. 148 (1970), 293–314. Zbl0224.17013MR0271181DOI10.1090/S0002-9947-1970-0271181-3
  18. 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. Zbl0765.17003MR1175866

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.