Degenerations of Filippov algebras

Abstract

Yury Volkov (wolf86 666@list.ru).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.

The geometric classification of nilpotent ℭ𝔇-algebras

“…, n})-commutative n-ary algebras we will called (a, c)commutative n-ary algebras. The geometric study of varieties of n-ary algebras defined by a family of polynomial identites has been started in [23]. Hence, we have an obvious open question.…”

“…The geometric study of varieties of n-ary algebras defined by a family of polynomial identites has been started in [23]. Hence, we have an obvious open question.…”

The algebraic and geometric classification of nilpotent left-symmetric algebras

“…Another interesting direction in the classification of algebras is the geometric classification. There are many results related to the geometric classification of Jordan, Lie, Leibniz, Zinbiel and many other algebras [5,8,10,11,21,22,26,27,30,32,33,37,40]. An algebraic classification of complex 3-dimensional left-symmetric algebras is given in [2].…”

$n$-ary algebras of the first level

“…Papers [1,10] concern this topic but do not give any complete result. Recently in the paper [8] the degenerations in the variety of (n + 1)-dimensional Filippov (n-Lie) algebras were described. The main tools for studying degenerations in the current paper will be taken from [8].…”

“…Recently in the paper [8] the degenerations in the variety of (n + 1)-dimensional Filippov (n-Lie) algebras were described. The main tools for studying degenerations in the current paper will be taken from [8]. In fact, these tools are generalizations of methods for binary algebras given in [2,11].…”