no. 2
Standard Subalgebras of Semisimple Lie Algebras and Computer-Aided for Enumeration
Es Saadi, B.1 ; Khakimdjanov, Yu.1 ; Makhlouf, A.1
1 Université de Haute Alsace Laboratoire de Mathématiques et Applications 4, rue des Frères Lumière 68093 Mulhouse Cedex FRANCE
Annales mathématiques Blaise Pascal, Tome 10 (2003) no. 2, pp. 315-326.

The aim of this work is to enumerate the standard subalgebras of a semisimple Lie algebra. The computations are based on the approach developed by Yu. Khakimdjanov in 1974. In this paper, we give a general formula for the number of standard subalgebras not necessarly nilpotent of a semisimple Lie algebra of type A p and the exceptional semisimple Lie algebras. With computer aided, we enumerate this number for the other types of small rank. Therefore, We deduce the number in the nilpotent case and describe a family of complete nilpotent standard subalgebras, these algebras are the nilradical of their normalizer.

DOI : 10.5802/ambp.180
Es Saadi, B. 1 ; Khakimdjanov, Yu. 1 ; Makhlouf, A. 1

1 Université de Haute Alsace Laboratoire de Mathématiques et Applications 4, rue des Frères Lumière 68093 Mulhouse Cedex FRANCE 
@article{AMBP_2003__10_2_315_0,
     author = {Es Saadi, B. and Khakimdjanov, Yu. and Makhlouf, A.},
     title = {Standard {Subalgebras} of {Semisimple} {Lie} {Algebras} and {Computer-Aided} for {Enumeration}},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {315--326},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {10},
     number = {2},
     year = {2003},
     doi = {10.5802/ambp.180},
     zbl = {1107.17005},
     mrnumber = {2031275},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/ambp.180/}
}
TY  - JOUR
AU  - Es Saadi, B.
AU  - Khakimdjanov, Yu.
AU  - Makhlouf, A.
TI  - Standard Subalgebras of Semisimple Lie Algebras and Computer-Aided for Enumeration
JO  - Annales mathématiques Blaise Pascal
PY  - 2003
SP  - 315
EP  - 326
VL  - 10
IS  - 2
PB  - Annales mathématiques Blaise Pascal
UR  - http://www.numdam.org/articles/10.5802/ambp.180/
DO  - 10.5802/ambp.180
LA  - en
ID  - AMBP_2003__10_2_315_0
ER  - 
%0 Journal Article
%A Es Saadi, B.
%A Khakimdjanov, Yu.
%A Makhlouf, A.
%T Standard Subalgebras of Semisimple Lie Algebras and Computer-Aided for Enumeration
%J Annales mathématiques Blaise Pascal
%D 2003
%P 315-326
%V 10
%N 2
%I Annales mathématiques Blaise Pascal
%U http://www.numdam.org/articles/10.5802/ambp.180/
%R 10.5802/ambp.180
%G en
%F AMBP_2003__10_2_315_0
Es Saadi, B.; Khakimdjanov, Yu.; Makhlouf, A. Standard Subalgebras of Semisimple Lie Algebras and Computer-Aided for Enumeration. Annales mathématiques Blaise Pascal, Tome 10 (2003) no. 2, pp. 315-326. doi : 10.5802/ambp.180. http://www.numdam.org/articles/10.5802/ambp.180/

[1] Cellini, P.; Papi, P. Ad-nilpotent ideals of a Borel subalgebras, Journal of algebra, Volume 225 (2000) | DOI | MR | Zbl

[2] Favre, G.; Santharoubane, L. Nilpotent Lie algebras of classical type, Journal of algebra, Volume 202 (1998) | DOI | MR | Zbl

[3] Gurevich, G. B. Standard Lie algebras, Math Sbornik (1954) | MR

[4] Khakimdjanov, Y. Standard subalgebras of reductive Lie algebras, Moscow University Mathematics Bulletin, Volume 29 (1974) | MR | Zbl

[5] Orsina, L.; Papi, P. Enumeration of ad-nilpotent ideals of a Borel subalgebras in type A by class nilpotence, C.R.Acad.Sci.Paris Seri I Math, Volume 330 (2000) | MR | Zbl

Cité par Sources :