Hochschild products and global non-abelian cohomology for algebras. Applications

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• 作者：A.L. Agore^a ; ^b ; ^{ana.agore@vub.ac.be" class="auth_mail" title="E-mail the corresponding author} ; ^{ana.agore@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; G. Militaru^c ; ^{gigel.militaru@fmi.unibuc.ro" class="auth_mail" title="E-mail the corresponding author} ; ^{gigel.militaru@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：16D70 ; 16S70 ; 16E40
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：221
• 期：2
• 页码：366-392
• 全文大小：663 K

文摘

Let A be a unital associative algebra over a field k, E   a vector space and π:E→A$\pi :E\to A$ a surjective linear map with V=Ker(π)$V=\mathrm{Ker}(\pi )$. All algebra structures on E   such that π:E→A$\pi :E\to A$ becomes an algebra map are described and classified by an explicitly constructed global cohomological type object $\mathbb{G}{\mathbb{H}}^2(A,V)$. Any such algebra is isomorphic to a Hochschild product A⋆V$A\star V$, an algebra introduced as a generalization of a classical construction. We prove that $\mathbb{G}{\mathbb{H}}^2(A,V)$ is the coproduct of all non-abelian cohomologies 19bb6365a5">${\mathbb{H}}^2(A,(V,\cdot ))$. The key object $\mathbb{G}{\mathbb{H}}^2(A,k)$ responsible for the classification of all co-flag algebras is computed. All Hochschild products 119ea9" title="Click to view the MathML source">A⋆k$A\star k$ are also classified and the automorphism groups Aut_Alg(A⋆k)${\mathrm{Aut}}_{\mathrm{Alg}}(A\star k)$ are fully determined as subgroups of a semidirect product $A^{⁎}⋉(k^{⁎}\times {\mathrm{Aut}}_{\mathrm{Alg}}(A))$ of groups. Several examples are given as well as applications to the theory of supersolvable coalgebras or Poisson algebras. In particular, for a given Poisson algebra P, all Poisson algebras having a Poisson algebra surjection on P with a 1-dimensional kernel are described and classified.