Define N ± as in (1) Moreover , we assume that there exists U + ? End, Then one has N = U N U ?1 ,
Then, m must be even (obviously, it happens if ? = ?1), m = 2n We decompose C = S + N into semisimple and nilpotent parts, S, N ? g ? by its Jordan decomposition. It is clear that S is invertible and ? ? ? if and only if ?? ? ? where ? is the spectrum of S. Also, b(? ) = b(?? ), for all ? ? ? with multiplicity b(? ). Since N and S commute, we have N(V ±? ) ? V ±? where V ? is the eigenspace of S corresponding to ? ? ?, Consider References [BBB] I. Bajo, S. Benayadi, and M. Bordemann, Generalized double extension and descriptions of quadratic Lie superalgebras, p.37 ,
Double extension of quadratic lie superalgebras, Communications in Algebra, vol.716, issue.2, pp.67-88, 1999. ,
DOI : 10.1080/00927879908826421
The graded Lie algebra structure of Lie superalgebra deformation theory, Lett. Math. Phys, vol.18, issue.9, pp.307-313, 1989. ,
Socle and some invariants of quadratic Lie superalgebras, Journal of Algebra, vol.261, issue.2, pp.245-291, 2003. ,
DOI : 10.1016/S0021-8693(02)00682-8
Nondegenerate invariant bilinear forms on nonassociative algebras, ?2 [Bou59] N. Bourbaki, ´ Eléments de Mathématiques. Algèbre, Formes sesquilinéaires et formes quadratiques, pp.151-201, 1959. ,
Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand-Reinhold Mathematics Series, pp.37-40, 1993. ,
A New Invariant of Quadratic Lie Algebras, Algebras and Representation Theory, vol.17, issue.3, pp.1163-1203, 2012. ,
DOI : 10.1007/s10468-011-9284-4
URL : https://hal.archives-ouvertes.fr/tel-00673991
Dominance Over The Classical Groups, The Annals of Mathematics, vol.74, issue.3, pp.532-569, 1961. ,
DOI : 10.2307/1970297
Nouvelles structures de Nambu et super-théorème d'Amitsur-Levizki, pp.8876-8885, 2004. ,
Conjugacy classes in semisimple algebraic groups, Mathematical Surveys and Monographs, vol.43, p.37, 1995. ,
DOI : 10.1090/surv/043
Infinite-dimensional Lie algebras, pp.2-17, 1985. ,
Alg??bres de Lie et produit scalaire invariant, Annales scientifiques de l'??cole normale sup??rieure, vol.18, issue.3, pp.553-561, 1985. ,
DOI : 10.24033/asens.1496
Hochschild Cohomology and Deformations of Clifford-Weyl Algebras, 27 pp. ?3, 2009. ,
DOI : 10.3842/SIGMA.2009.028
URL : https://hal.archives-ouvertes.fr/hal-00438865
Cohomology and deformations in graded Lie algebras, Bulletin of the American Mathematical Society, vol.72, issue.1, pp.1-29, 1966. ,
DOI : 10.1090/S0002-9904-1966-11401-5
New Applications of Graded Lie Algebras to Lie Algebras, Generalized Lie Algebras, and Cohomology, J. Lie Theory, vol.17, issue.3, pp.633-668, 2007. ,
URL : https://hal.archives-ouvertes.fr/hal-00438859
The Theory of Lie Superalgebras, Lecture Notes in Mathematics, vol.716, p.4, 1979. ,
DOI : 10.1007/BFb0070929