The algebraic and geometric classification of nilpotent Novikov algebras

Abstract

This paper is devoted to give the complete algebraic and geometric classification of 4dimensional nilpotent Novikov algebras over C.

One-generated nilpotent Novikov algebras

“…Low dimensional one-generated nilpotent Novikov algebras. Thanks to [35] we have the algebraic classification of 2-, 3and 4dimensional and 5-dimensional with 2-dimensional annihilator one-generated nilpotent algebras. Let us give a list if 1-, 2-, 3and 4-dimensional one-generated nilpotent Novikov algebras.…”

“…Also, the simple Novikov algebras were described in infinite-dimensional case and over fields of positive characteristic [44,46,48]. The algebraic classification of 3-dimensional Novikov algebras was given in [4], and for some classes of 4-dimensional algebras, it was given in [7]; the geometric classification of 3-dimensional Novikov algebras was given in [5] and of 4-dimensional nilpotent Novikov algebras in [35]. Many other pure algebraic properties were studied in a series of papers by Dzhumadildaev [21][22][23][24].…”

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 3 e 4 = e 6 ; A 31 : e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 1 e 4 = e 5 , e 2 e 3 = e 6 , e 2 e 5 = e 6 , e 3 e 4 = e 6 ; A 32 (α) : e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 1 e 4 = e 5 , e 2 e 3 = αe 6 , e 2 e 5 = e 6 , e 4 e 5 = e 6 ; A 33 : e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 1 e 4 = e 5 , e 2 e 3 = e 6 , e 4 e 5 = e 6 ; A 34 : e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 1 e 4 = e 5 , e 2 e 4 = e 6 , e 2 e 5 = −e 6 , e 3 e 4 = e 6 ; A 35 (α) : e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 1 e 4 = e 5 , e 2 e 4 = αe 6 , e 2 e 5 = e 6 , e 3 e 5 = e 6 ; A 36 : e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 1 e 4 = e 5 , e 2 e 4 = e 6 , e 3 e 5 = e 6 ; A 37 (α) : e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 1 e 4 = e 5 , e 2 e 5 = αe 6 , e 3 e 4 = e 6 ; A 38 : e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 1 e 4 = e 5 , e 3 e 5 = e 6 ; A 39 : e 1 e 2 = e 3 , e 1 e 3 = e 4 , e 1 e 4 = e 5 , e 4 e 5 = e 6 .…”

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.…”