The Algebraic and Geometric Classification of Nilpotent Assosymmetric Algebras

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

“…: 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.…”

“…These algebras are of big interest, since the closures of their orbits under the action of the generalized linear group form irreducible components of the variety under consideration (with respect to the Zariski topology). For example, the rigid algebras in the varieties of all 4-dimensional Leibniz algebras [24], all 4-dimensional nilpotent Novikov algebras [26], all 4-dimensional nilpotent bicommutative algebras [27], all 4-dimensional nilpotent assosymmetric algebras [23], all 6-dimensional nilpotent binary Lie algebras [1], and some other has been classified. There are fewer works in which the full information about degenerations has been found for some variety of algebras.…”