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 \(sl(2,\mathbb{C})\otimes\mathbb{C}^n\).


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


Zbl 0734.17008
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