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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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