×

zbMATH — the first resource for mathematics

Sur certaines classes d’algèbres de Lie rigides. (On some classes of rigid Lie algebras). (French) Zbl 0559.17010
This work proposes minorations of the number of open orbits (solvable and non-solvable) in the variety \(L_ m\) of m-dimensional Lie algebras. These minorants increase very fastly with m in spite of the restrictive conditions imposed to the tested family of Lie algebras. We give then a series of solvable and rigid Lie algebras which do not annihilate the second adjoint cohomology group. The quadratic application of Rim is zero for this series. This implies that the variety \(L_ m\) is not reduced (if \(m\geq 13)\) for its scheme structure defined by the Jacobi rules.

MSC:
17B99 Lie algebras and Lie superalgebras
17B30 Solvable, nilpotent (super)algebras
17B05 Structure theory for Lie algebras and superalgebras
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Bourbaki, N.: Groupes et algèbres de Lie. Chap. I. Paris: Hermann 1971 · Zbl 0213.04103
[2] Bratzlavsky, F.: Sur les algèbres admettant un, tore d’automorphisme donné. J. Algebra30, 305-316 (1974) · Zbl 0287.16021
[3] Carles, R.: Sur certaines classes d’orbites ouvertes dans les variétés d’algèbres de Lie. C.R. Acad. Sci. Paris, Ser. I293, 545-547 (1981) · Zbl 0481.17005
[4] Carles, R.: Sur la structure des algèbres de Lie rigides. Ann. Inst. Fourier34, 65-82 (1984) · Zbl 0519.17004
[5] Carles, R., Diakité, Y.: Sur les variétés d’algèbres de Lie de dimension ?7. J. Algebra91, 53-63 (1984) · Zbl 0546.17006
[6] Favre, M.: Algèbres de Lie complètes. Thèse Lausanne (1974)
[7] Goze, M.: Etude locale de la variété des lois d’algèbres de Lie. Thèse, Mulhouse (1982) · Zbl 0573.58024
[8] Leger, G., Luks, E.: Cohomology theorems for borel-like solvable Lie algebras in arbitrary characteristic. Can. J. Math.24, 1019-1026 (1972) · Zbl 0272.17004
[9] Rauch, G.: Effacement et déformation. Ann. Inst. Fourier22, 239-269 (1972) · Zbl 0219.17006
[10] Richardson, R.W., Jr.: On the rigidity of semi-direct products of Lie algebras. Pac. J. Math.22, 339-344 (1967) · Zbl 0166.30301
[11] Rim, D.S.: Deformation of transitive Lie algebras. Ann. Math. 339-357 (1966) · Zbl 0136.43104
[12] Skjelbred, T., Sund, T.: Sur la classification des algèbres de Lie nilpotentes. C.R. Acad. Sci. Paris Ser. I286, 241-242 (1978) · Zbl 0375.17006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.