The algebraic and geometric classification of nilpotent anticommutative algebras

Abstract

We give algebraic and geometric classifications of 6-dimensional complex nilpotent anticommutative algebras. Specifically, we find that, up to isomorphism, there are 14 one-parameter families of 6-dimensional nilpotent anticommutative algebras, complemented by 130 additional isomorphism classes. The corresponding geometric variety is irreducible and determined by the Zariski closure of a one-parameter family of algebras. In particular, there are no rigid 6-dimensional complex nilpotent anticommutative algebras.

The geometric classification of nilpotent ℭ𝔇-algebras

“…Let n ≥ 6. Then dim T ∈Tn O(T ) = (n−2)(n 2 +2n+3) In case n = 6, the statement follows from Example 28 and[18, Thm. 2].…”

“…The base case n = 6 has already been proved in Example 28 and [18]. Denote by γ k ij the structure constants of N in (e i ) n+1…”

“…It follows that A(α, β) ≃ A 82 (−α/β 2 ), where the family A 82 (γ), for γ ∈ C * , was defined in [18,Thm. 1] and shown to be generic in the variety of 6-dimensional complex nilpotent anticommutative algebras in [18,Thm. 2].…”

The variety of dual mock-Lie algebras

“…Thanks to [20], we have the classification of all 6-dimensional nilpotent anticommutative algebras and choosing only dual mock-Lie algebras from the list of algebras presented in [20] we have the classification of all low dimensional dual mock-Lie algebras. By the straightforward verification, it follows that only…”

The algebraic classification of nilpotent commutative algebras

“…The algebraic classification (up to isomorphism) of algebras of dimension n from a certain variety defined by a certain family of polynomial identities is a classic problem in the theory of non-associative algebras. There are many results related to the algebraic classification of small-dimensional algebras in many varieties of non-associative algebras [11,12,2,3,4,6,13,9,16]. So, algebraic classifications of 2-dimensional algebras [16,19], 3-dimensional evolution algebras [1], 3-dimensional anticommutative algebras [17], 4-dimensional division algebras [7,5], 4-dimensional nilpotent algebras [13] and 6-dimensional anticommutative nilpotent algebras [12] have been given.…”