Crossed products of $4$-algebras. Applications

Abstract

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 $\pi :E\to A$ a surjective linear map with $V=\mathrm{Ker}(\pi )$. 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 $\mathbb{G}{\mathbb{H}}^2(A,V)$. Any such $4$-algebra is isomorphic to a crossed product $V\#A$ and $\mathbb{G}{\mathbb{H}}^2(A,V)$ is a coproduct, over all $4$-algebras structures ${\cdot }_V$ on $V$, of all non-abelian cohomologies ${\mathbb{H}}_{\mathrm{nab}}^2(A,(V,{\cdot }_V))$, 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 $\mathrm{Gal}(V\#A/V)$ of the extension $V\hookrightarrow V\#A$ is described.

The final version will appear in J. Algebra. arXiv admin note: text overlap with arXiv:1507.08146, arXiv:1503.05364

References in corpus (2)

• A new invariant for finite dimensional Leibniz/Lie algebras
• The Finite Duals of Affine Prime Regular Hopf Algebras of GK-Dimension One

Cited by in corpus (3)

• Universal Coacting Hopf algebra of a Finite dimensional Lie-Yamaguti algebra
• Universal Coacting Hopf Algebra of a finite-dimensional Algebra over an Operad
• Automorphism group of Suzuki's Hopf algebra