## Deformations in the schemes defined by Jacobi identities. (Déformations dans les schémas définis par les identités de Jacobi.)(French)Zbl 0919.17015

Let $$\Phi_0$$ be a Lie algebra law on the vector space $$k^n$$ $$(k$$ is an algebraically closed field of characteristic zero). The author works on the category of deformations of $$\Phi_0$$ parametrized by a local ring $$A$$, understood as morphisms $${\mathcal O}\to A$$ where $${\mathcal O}$$ is the local ring at the point $$\Phi_0$$ in the scheme defined by antisymmetric and Jacobi identities. The definition contains deformations in the sense of Gerstenhaber [R. Carles, C.R. Acad. Sci., Paris, Sér. I 312, 671-674 (1991; Zbl 0734.17008)]. If $$A$$ is complete, the author shows that each deformation – up to an equivalence – has certain structure constants fixed and constant values. Deformations expressed with parameters which are running over orbits (under the canonical action of $$GL(n,k))$$ distinct from the orbit of $$\Phi_0$$ are studied. In particular, the number of the essential parameters is calculated. Finally, the author examines the case of the complex Lie algebra $$sl(2,\mathbb{C})\otimes\mathbb{C}^n$$.

### MSC:

 17B56 Cohomology of Lie (super)algebras 14D15 Formal methods and deformations in algebraic geometry

Zbl 0734.17008
Full Text:

### References:

 [1] Bourbaki, N., Algèbre, chap. 4 à 7, Masson (1981) · Zbl 0498.12001 [2] Bratzlavsky, F., Sur les algèbres admettant un tore d’automorphismes donné . J.of Alg.30 , 305-316 (1974). · Zbl 0287.16021 [3] Carles, R., Sur certaines classes d’algèbres de Lie rigides. Math.ann.272, 477-488 (1985). · Zbl 0559.17010 [4] Carles, R., Déformations et éléments nilpotents dans les schémas définis par les identités de Jacobi. C.R.A.S.312 , 671-674 (1991). · Zbl 0734.17008 [5] Carles, R., Diakite, Y., Sur les variétés d’algèbres de Lie de dimension ≤7. J.of Alg.91, 53-63 (1984). · Zbl 0546.17006 [6] Gerstenhaber, M., The cohomology structure of an associative ring. Ann.of Math.78, 2, 267-288 (1963). · Zbl 0131.27302 [7] Gerstenhaber, M., On the deformations of rings and algebras. Ann.of Math.79, 59-103 (1964). · Zbl 0123.03101 [8] Gerstenhaber, M., Schack, S.D., Relative Hochschild cohomology, rigid algebras, and the Bockstein. J. of Pure and Appl. Alg.43 , 53-74 (1986). · Zbl 0603.16021 [9] Inonu, E., Wigner, E.P., On the contraction of groups and their representation, Proc.Nat.Acad.Sci.39, 510-526 (1953). · Zbl 0050.02601 [10] Kostant, B., The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math.81, 973-1032 (1959). · Zbl 0099.25603 [11] Nijenhuis, A., Richardson, R.W., Cohomology and deformations in graded Lie Algebras, Bull.Amer.Math.Soc.72, 1 (1966). · Zbl 0136.30502 [12] Rauch, G., Effacement et déformation. Ann.Inst.Fourier22, 239-269 (1972). · Zbl 0219.17006 [13] Richardson, R.W., The rigidity of semi-direct products of Lie algebras. Pacific J.of Math.22, 339-344 (1967). · Zbl 0166.30301
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.