The algebraic classification of nilpotent Tortkara algebras

Abstract

We classify all 6-dimensional nilpotent Tortkara algebras over C.

The algebraic and geometric classification of nilpotent anticommutative algebras

“…All of the above, combined with the algebraic classification of 6-dimensional nilpotent Tortkara and Malcev algebras in [22] and [32], respectively, yields our main result of this section. In the present work we use the methods applied to Lie algebras in [9,27,28,48].…”

“…The class of anticommutative algebras includes all Malcev (in particular, all Lie) and all Tortkara algebras. Concerning the latter, the algebraic and geometric classifications of 6-dimensional nilpotent Tortkara algebras have been completed in [22] and [23], respectively. We will rely on this work; in particular, we will be able to proceed in our algebraic and geometric classifications modulo the class of Tortkara algebras.…”

“…We will rely on this work; in particular, we will be able to proceed in our algebraic and geometric classifications modulo the class of Tortkara algebras. We will also rely on the classification of 4-dimensional nilpotent anticommutative algebras in [10], 5-dimensional nilpotent anticommutative algebras in [17], 6-dimensional nilpotent Malcev algebras in [32] and 6-dimensional nilpotent Tortkara algebras in [22][23][24].…”

“…An important such class is that of Tortkara algebras. These are anticommutative algebras satisfying the identity The algebraic classification of all 6-dimensional nilpotent Tortkara algebras was completed in [22] and their geometric classification was obtained in [23]. Therefore, our classification of anticommutative algebras will be carried out modulo the class of Tortkara algebras.…”

The geometric classification of nilpotent Tortkara algebras

“…There are many results related to both the algebraic and geometric classification of small dimensional algebras in the varieties of Jordan, Lie, Leibniz and Zinbiel algebras; for algebraic results see, for example, [1,11,18,19,23,[25][26][27]30]; for geometric results see, for example, [1, 3-6, 8, 10, 11, 19-31, 34]. Here we give a geometric classification of 6-dimensional nilpotent Tortkara algebras over C. Our main result is Theorem 3 which describes the rigid algebras in this variety.…”