The algebraic and geometric classification of Zinbiel algebras

The algebraic and geometric classification of transposed Poisson algebras

“…The study of degenerations of algebras is very rich and closely related to deformation theory, in the sense of Gerstenhaber [20]. The geometric classification is given for many varieties of non-associative algebras (see, for example, [2,3,8,15,16,16,22,25] and references therein). Degenerations have also been used to study a level of complexity of an algebra [21,36].…”

“…β i 33 e 2 , ϕ i (e 3 ) = β i 31 e 1 + β i 32 e 2 + β i 33 e 3 and β1 33 e 2 = ϕ 1 (e 2 ) = ϕ 2 (e 1 ) = β 2 33 e 1 , β1 31 e 1 + β 1 32 e 2 + β 1 33 e 3 = ϕ 1 (e 3 ) = ϕ 3 (e 1 ) = β 3 33 e 1 , β3 33e 2 = ϕ 2 (e 3 ) = ϕ 3 (e 2 ) = β 2 31 e 1 + β 2 32 e 2 + β 2 33 e 3 .Hence, the commutative multiplication − • − is defined bye 1 • e 3 = β 3 33 e 1 , e 2 • e 3 = β3 33 e 2 , e 3 • e 3 = β3 31 e 1 + β 3 32 e 2 + β 3 33 e 3 . Through straightforward calculations, it is possible to conclude that − • − is associative too.…”

“…Through straightforward calculations, it is possible to conclude that − • − is associative too. We are interested in (β3 31 , β 3 32 , β 3 33 ) = 0. Let us now separate all non-isomorphic cases.…”

The algebraic classification and degenerations of nilpotent Poisson algebras

“…The algebraic classification (up to isomorphism) of algebras of dimension n of a certain variety defined by a family of polynomial identities is a classic problem in the theory of non-associative algebras, see [1][2][3][4][5]8,10,20,21,26,27,29,39,41,44,45,48,51]. There are many results related to the algebraic classification of small-dimensional algebras in different varieties of associative and non-associative algebras.…”