The geometric classification of nilpotent Tortkara algebras

Abstract

We give a geometric classification of all 6-dimensional nilpotent Tortkara algebras over C.

The variety of dual mock-Lie algebras

“…There are many results related to the algebraic and geometric classification of low dimensional algebras in the varieties of Jordan, Lie, Leibniz and Zinbiel algebras; for the algebraic classification see, for example, [1], [6], [7], [8], [9], [19], [22]; for the geometric classification and descriptions of degenerations see, for example, [1], [2], [3], [5], [11], [12], [13], [16], [17], [19], [21], [22], [23], [26]. Here we give the algebraic and geometric classification of complex dual mock-Lie algebras of small dimensions.…”

The algebraic and geometric classification of nilpotent anticommutative algebras

“…For example, rigid algebras have been classified in the following varieties: 4-dimensional Leibniz algebras [34], 4-dimensional nilpotent Novikov algebras [36], 4-dimensional nilpotent bicommutative algebras [39], 6-dimensional nilpotent binary Lie algebras [1], 6-dimensional nilpotent Tortkara algebras [23]. There are fewer works in which the full information about degenerations has been given for some variety of algebras.…”

“…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.…”

“…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.…”

“…Proof. Thanks to [23] the algebras T 10 , T 17 and T 19 (see Appendix B) are rigid in the variety of 6-dimensional nilpotent Tortkara algebras. These algebras and the remaining 6-dimensional nilpotent anticommutative algebras degenerate from A 82 α , as shown in the table below.…”