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 846&_mathId=si1.gif&_user=111111111&_pii=S0022404916300846&_rdoc=1&_issn=00224049&md5=950723cc4f86a0c5bae3c7dda614bdbd" title="Click to view the MathML source">π:E→A$\pi :E\to A$ a surjective linear map with 846&_mathId=si146.gif&_user=111111111&_pii=S0022404916300846&_rdoc=1&_issn=00224049&md5=d47e27c30971d1948fb178d009b290f1" title="Click to view the MathML source">V=Ker(π)$V=\mathrm{Ker}(\pi )$. All algebra structures on E   such that 846&_mathId=si1.gif&_user=111111111&_pii=S0022404916300846&_rdoc=1&_issn=00224049&md5=950723cc4f86a0c5bae3c7dda614bdbd" title="Click to view the MathML source">π:E→A$\pi :E\to A$ becomes an algebra map are described and classified by an explicitly constructed global cohomological type object 846&_mathId=si3.gif&_user=111111111&_pii=S0022404916300846&_rdoc=1&_issn=00224049&md5=20a827da2224eb798b41a806189d7cd3">846-si3.gif">$\mathbb{G}{\mathbb{H}}^2(A,V)$. Any such algebra is isomorphic to a Hochschild product 846&_mathId=si111.gif&_user=111111111&_pii=S0022404916300846&_rdoc=1&_issn=00224049&md5=774239def1031e83c3d5db38a448633b" title="Click to view the MathML source">A⋆V$A\star V$, an algebra introduced as a generalization of a classical construction. We prove that 846&_mathId=si3.gif&_user=111111111&_pii=S0022404916300846&_rdoc=1&_issn=00224049&md5=20a827da2224eb798b41a806189d7cd3">846-si3.gif">$\mathbb{G}{\mathbb{H}}^2(A,V)$ is the coproduct of all non-abelian cohomologies 846&_mathId=si6.gif&_user=111111111&_pii=S0022404916300846&_rdoc=1&_issn=00224049&md5=b6c82aff1a59034cbdc13b19bb6365a5">846-si6.gif">${\mathbb{H}}^2(A,(V,\cdot ))$. The key object 846&_mathId=si278.gif&_user=111111111&_pii=S0022404916300846&_rdoc=1&_issn=00224049&md5=78a3455e1d9c08abb0b87f86e08af52b">846-si278.gif">$\mathbb{G}{\mathbb{H}}^2(A,k)$ responsible for the classification of all co-flag algebras is computed. All Hochschild products 846&_mathId=si298.gif&_user=111111111&_pii=S0022404916300846&_rdoc=1&_issn=00224049&md5=a4c10dd5a52ed4e465515fa3e7119ea9" title="Click to view the MathML source">A⋆k$A\star k$ are also classified and the automorphism groups 846&_mathId=si9.gif&_user=111111111&_pii=S0022404916300846&_rdoc=1&_issn=00224049&md5=3fe379aff930185d34df3096e4bbb487" title="Click to view the MathML source">Aut_Alg(A⋆k)${\mathrm{Aut}}_{\mathrm{Alg}}(A\star k)$ are fully determined as subgroups of a semidirect product 846&_mathId=si10.gif&_user=111111111&_pii=S0022404916300846&_rdoc=1&_issn=00224049&md5=9e2d7085e46daead01f25886808957cc">846-si10.gif">$A^{⁎}⋉(k^{⁎}\times {\mathrm{Aut}}_{\mathrm{Alg}}(A))$ 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.