Crossed products of $4$-algebras. Applications

A $4$-algebra is a commutative algebra $A$ over a field $k$ such that $(a^2{)}^2=0$, for all $a\in A$. We have proved recently \cite{Mil} that $4$-algebras play a prominent role in the classification of finite dimensional Bernstein algebras. Let $A$ be a $4$-algebra, $E$ a vector space and $\pi : E \to A$asurjectivelinearmapwith$V = {\rm Ker} (\pi)$\mathrm{.}All$4$-algebra structures on $E$ such that $\pi :E\to A$ is an algebra map are described and classified by a global cohomological object ${\mathbb G} {\mathbb H}^{2} , (A, , V)$\mathrm{.}Anysuch$4$-algebraisisomorphictoacrossedproduct$V # A$and${\mathbb G} {\mathbb H}^{2} , (A, , V)$isacoproduct,overall$4$-algebrasstructures$\cdot_V$on$V$,ofallnon-abeliancohomologies${\mathbb H}^{2}{\rm nab} , \bigl(A, , (V, , \cdot{V} )\bigl)$, which are the classifying objects for all extensions of $A$ by $V$. Several applications and examples are provided: in particular, ${\mathbb G} {\mathbb H}^{2} , (A, , k)$and${\mathbb G} {\mathbb H}^{2} , (k, , V)$ are explicitly computed and the Galois group $\mathrm{G}\mathrm{a}\mathrm{l}(V\mathrm{\#}A\mathrm{/}V)$ of the extension $V \hookrightarrow V # A$ is described.