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Crossed products of $4$-algebras. Applications
- Source :
- J. Algebra, Vol. 614 (2023), 251-270
- Publication Year :
- 2022
-
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 $V = {\rm Ker} (\pi)$. 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 ${\mathbb H}^{2}_{\rm nab} \, \bigl(A, \, (V, \, \cdot_{V} )\bigl)$, 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 ${\rm Gal} \, (V \# A/ V )$ of the extension $V \hookrightarrow V \# A$ is described.<br />Comment: The final version will appear in J. Algebra. arXiv admin note: text overlap with arXiv:1507.08146, arXiv:1503.05364
- Subjects :
- Mathematics - Rings and Algebras
17A60, 17A30, 17D92
Subjects
Details
- Database :
- arXiv
- Journal :
- J. Algebra, Vol. 614 (2023), 251-270
- Publication Type :
- Report
- Accession number :
- edsarx.2204.09474
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1007/s00025-021-01509-z