Degenerations of nilpotent Lie algebras

Abstract: In this paper we study degenerations of nilpotent Lie algebras. If $\pt8{\la,\mu}$ are two points in the variety of nilpotent Lie algebras, then $\pt8{\la}$ is said to degenerate to $\pt8{\mu}$, $\pt8{\la \ra_{deg} \mu}$, if $\pt8{\mu}$ lies in the Zariski closure of the orbit of $\pt8{\la}$. It is known that all degenerations of nilpotent Lie algebras of dimension $\pt8{n <7}$ 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 $\pt8{n \ge 7}$ there exist examples of degenerations of nilpotent Lie algebras which cannot be realized via a one-parameter subgroup.

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