Solvability of a Commutative Algebra which Satisfies (x²)² = 0

On the structure and classification of Bernstein algebras

“…) and v 0 := ψ 1 (0, 1) ∈ W . Then, ψ 1 (x, α) = f (x)+αv 0 and thus any linear map ψ : V × k → W × k is uniquely determined by a pair (v 0 , f ) ∈ W × Hom k (V, W ) via the formula (14). Now, by a straightforward computation we can prove that such a map ψ = ψ (v 0 ,f…”

“…Remark 2.2. The class of 4-algebras were introduced and studied in [14] (see also [9]) where it was proved that any 4-algebra of dimension ≤ 7 is solvable and it was conjectured that any finite dimensional 4-algebra is solvable. If (A, ω) is a Bernstein algebra, then Ker(ω) is a 4-algebra [22].…”

“…In this context, we recall the conjecture formulated by Guzzo-Benh [14] that was proved in the affirmative sense up to dimension 7: is any finite dimensional 4-algebra solvable?…”

Crossed products of $4$-algebras. Applications

“…for 4-algebras are defined in the obvious way. The class of 4-algebras were studied before in [14] where it was proved that any 4-algebra of dimension ≤ 7 is solvable and it was conjectured that any finite dimensional 4-algebra is solvable. Any vector space V is a 4-algebra with the trivial multiplication x • y := 0, for all x, y ∈ V : we call this algebra abelian and it will be denoted by V 0 .…”

“…Conversely, if A is a 4-algebra and Ω = Ω 2 ∈ End k (A) is a Bernstein operator on A, then A × k has a canonical structure of Bernstein algebra with the barideal A (for details see [20,Proposition 2.3]). Linearizing several times the second compatibility of (2) we obtain that in a 4-algebra the following relations hold [14]:…”