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A comparison of deformations and orbit closure. (English) Zbl 0719.17002

The paper has the purpose to point out similarities and differences in the two notions of deformations on the one hand side and orbit closure (called degeneration or contraction, too) on the other. The category the authors deal with most of the time are Lie algebras. In section 1 they start by generalizing Gerstenhaber’s definition of a formal deformation in correspondence with the viewpoint of orbit closure introduced by Grunewald and O’Halloran. It is shown that if the orbit is not Zariski closed, then a point in the closure always arises from a deformation.
In section 2 the existence of versal deformations and universal degenerations is studied and in section 3 the concept of rigidity.

MSC:

17B05 Structure theory for Lie algebras and superalgebras
14D15 Formal methods and deformations in algebraic geometry
17B56 Cohomology of Lie (super)algebras
20G99 Linear algebraic groups and related topics
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