Degenerations of binary Lie and nilpotent Malcev algebras

Abstract

Yury Volkov (wolf86 666@list.ru).Abstract. We describe degenerations of four-dimensional binary Lie algebras, and five-and six-dimensional nilpotent Malcev algebras over C. In particular, we describe all irreducible components of these varieties.

The geometric classification of nilpotent Tortkara algebras

“…2. THE GEOMETRIC CLASSIFICATION OF 6-DIMENSIONAL NILPOTENT TORTKARA ALGEBRAS The geometric classification of 6-dimensional nilpotent Tortkara algebras is based on the description of all degenerations of 6-dimensional nilpotent Malcev algebras [29]. Note that the algebras g 6 and g 8 from [29] are not Tortkara.…”

“…THE GEOMETRIC CLASSIFICATION OF 6-DIMENSIONAL NILPOTENT TORTKARA ALGEBRAS The geometric classification of 6-dimensional nilpotent Tortkara algebras is based on the description of all degenerations of 6-dimensional nilpotent Malcev algebras [29]. Note that the algebras g 6 and g 8 from [29] are not Tortkara. Hence, every 6-dimensional nilpotent Tortkara-Malcev algebra degenerates from one of the following algebras: g 5 : e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 1 e 4 = e 5 , e 1 e 5 = e 6 , e 2 e 3 = e 5 , e 2 e 4 = e 6 , M ǫ 6 : e 1 e 2 = e 3 , e 1 e 3 = e 5 , e 1 e 5 = e 6 , e 2 e 4 = ǫe 5 , e 3 e 4 = e 6 .…”

The algebraic and geometric classification of nilpotent anticommutative algebras

“…: e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 1 e 4 = e 5 , e 1 e 5 = e 6 , e 2 e 3 = e 5 , e 2 e 5 = e 6 , e 3 e 4 = −e 6 ; A 41 : e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 1 e 4 = e 5 , e 1 e 5 = e 6 , e 2 e 3 = e 5 , e 3 e 4 = e 6 ; A 42 : e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 1 e 4 = e 5 , e 2 e 3 = e 5 , e 2 e 4 = e 6 , e 3 e 5 = e 6 ; A 43 (α) : e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 1 e 4 = e 5 , e 2 e 3 = e 5 , e 2 e 5 = αe 6 , e 3 e 4 = e 6 ; A 44 : e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 1 e 4 = e 5 , e 2 e 3 = e 5 , e 3 e 5 = e 6 ; A 45 : e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 1 e 4 = e 5 , e 2 e 3 = e 5 , e 4 e 5 = e 6 .…”

Degenerations of nilpotent associative commutative algebras

“…There are fewer works in which the full information about degenerations was given for some variety of algebras. This problem was solved for 2-dimensional pre-Lie algebras [6], for 2-dimensional terminal algebras [9], for 3-dimensional Novikov algebras [7], for 3dimensional Jordan algebras [15], for 3-dimensional Jordan superalgebras [5], for 3-dimensional Leibniz and 3-dimensional anticommutative algebras [20], for 4-dimensional Lie algebras [8], for 4-dimensional Lie superalgebras [4], for 4-dimensional Zinbiel and 4-dimensional nilpotent Leibniz algebras [23], for 5-dimensional nilpotent Tortkara algebras [14], for 6-dimensional nilpotent Lie algebras [16,30], for 6dimensional nilpotent Malcev algebras [24], for 7-dimensional 2-step nilpotent Lie algebras [3], and for all 2-dimensional algebras [25]. Here we construct the graphs of primary degenerations for the variety of complex 5-dimensional nilpotent associative commutative algebras.…”