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Degenerations of nilpotent Lie algebras. (English) Zbl 1063.17009
Summary: In this paper we study degenerations of nilpotent Lie algebras. If $$\lambda,\mu$$ are two points in the variety of nilpotent Lie algebras, then $$\lambda$$ is said to degenerate to $$\mu$$, $$\lambda \rightarrow_{\text{deg}} \mu$$, if $$\mu$$ lies in the Zariski closure of the orbit of $$\lambda$$. It is known that all degenerations of nilpotent Lie algebras of dimension $$n <7$$ can be realized via a one-parameter subgroup. We construct degenerations between characteristically nilpotent filiform Lie algebras. As an application it follows that for any dimension $$n \geq 7$$ there exist examples of degenerations of nilpotent Lie algebras which cannot be realized via a one-parameter subgroup.

##### MSC:
 17B30 Solvable, nilpotent (super)algebras
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