The variety of dual mock-Lie algebras

Abstract

We classify all complex 7- and 8-dimensional dual mock-Lie algebras by the algebraic and geometric way. Also, we find all non-trivial complex 9-dimensional dual mock-Lie algebras.

Unified computational approach to nilpotent algebra classification problems

“…If for some i, j the product pr[e[i], e[j]] = 0 then drop it. For instance, pr [ e [ 1 ] , e [ 1 ] ] : = e [ 2 ] ; pr [ e [ 1 ] , e [ 2 ] ] : = e [ 3 ] ; pr [ e [ 2 ] , e [ 1 ] ] : = e [ 3 ] ;…”

“…Varieties of associative, alternative, Lie, Novikov, Jordan, assosymmetric, Leibniz, Zinbiel, and Tortkara algebras are some of the most studied varieties, see e.g. [1], [2], [6], [8], [15], [17] and references therein. As described in the next section, classification involves several steps each requiring to symbolically solve systems of polynomial equations.…”

One-generated nilpotent bicommutative algebras

“…After that, and basing on the method developed by Skjelbred and Sund, all non-Lie central extensions of all 4-dimensional Malcev algebras were described [24], all anticommutative central extensions of 3-dimensional anticommutative algebras [6], and all central extensions of 2-dimensional algebras [8]. Also, central extensions were used to give classifications of 4-dimensional nilpotent associative algebras [15], 5-dimensional nilpotent Jordan algebras [23], 5dimensional nilpotent restricted Lie algebras [13], 5-dimensional nilpotent associative commutative algebras [36], 6-dimensional nilpotent Lie algebras [12,14], 6-dimensional nilpotent Malcev algebras [25], 6-dimensional nilpotent anticommutative algebras [30], 8-dimensional dual Mock Lie algebras [11], and some others.…”

The algebraic classification of nilpotent ℭ𝔇-algebras

“…Using the same method, all non-Lie central extensions of all 4-dimensional Malcev algebras [21], all non-associative central extensions of all 3dimensional Jordan algebras [20], all anticommutative central extensions of 3-dimensional anticommutative algebras [5], all central extensions of 2-dimensional algebras [7] and some others were described. One can also look at the classification of 3-dimensional nilpotent algebras [16], 4-dimensional nilpotent associative algebras [13], 4-dimensional nilpotent Novikov algebras [27], 4-dimensional nilpotent bicommutative algebras [32], 4-dimensional nilpotent commutative algebras in [16], 4-dimensional nilpotent assosymmetric algebras in [25], 4-dimensional nilpotent noncommutative Jordan algebras in [26], 4-dimensional nilpotent terminal algebras [31], 5-dimensional nilpotent restricted Lie algebras [11], 5-dimensional nilpotent associative commutative algebras [34], 5-dimensional nilpotent Jordan algebras [19], 6-dimensional nilpotent Lie algebras [10,12], 6-dimensional nilpotent Malcev algebras [22], 6-dimensional nilpotent Tortkara algebras [17,18], 6-dimensional nilpotent binary Lie algebras [1], 6-dimensional nilpotent anticommutative CD-algebras [1], 6-dimensional nilpotent anticommutative algebras [29], 8-dimensional dual mock-Lie algebras [8].…”