Degenerations of Filippov algebras

TL;DR

This paper studies how Filippov (n-Lie) algebra structures on (n+1)-dimensional spaces can degenerate, classifies all such degenerations, and introduces criteria and invariants for understanding these processes.

Contribution

It provides a comprehensive classification of orbit closures and degenerations of Filippov algebras, introducing new invariants and criteria for degeneration.

Findings

Classification of all orbit closures in complex (n+1)-dimensional Filippov n-ary algebras

Development of trace invariants for degeneration analysis

Establishment of necessary criteria for algebra degenerations

Abstract

We consider the variety of Filippov ($n$-Lie) algebra structures on an $(n+1)$-dimensional vector space. The group $GL_n(K)$ acts on it, and we study the orbit closures with respect to the Zariski topology. This leads to the definition of Filippov algebra degenerations. We present some fundamental results on such degenerations, including trace invariants and necessary degeneration criteria. Finally, we classify all orbit closures in the variety of complex $(n+1)$-dimensional Filippov $n$-ary algebras.