Hochschild products and global non-abelian cohomology for algebras. Applications
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- 作者：A.L. Agore<sup>asup><sup> ; sup><sup>bsup><sup> ; sup><sup>ana.agore@vub.ac.be" class="auth_mail" title="E-mail the corresponding authorsup><sup> ; sup><sup>ana.agore@gmail.com" class="auth_mail" title="E-mail the corresponding authorsup> ; G. Militaru<sup>csup><sup> ; sup><sup>gigel.militaru@fmi.unibuc.ro" class="auth_mail" title="E-mail the corresponding authorsup><sup> ; sup><sup>gigel.militaru@gmail.com" class="auth_mail" title="E-mail the corresponding authorsup>
- 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 <span id="mmlsi1" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300846&_mathId=si1.gif&_user=111111111&_pii=S0022404916300846&_rdoc=1&_issn=00224049&md5=950723cc4f86a0c5bae3c7dda614bdbd" title="Click to view the MathML source">π:E→Aspan><span class="mathContainer hidden"><span class="mathCode">span>span>span> a surjective linear map with <span id="mmlsi146" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300846&_mathId=si146.gif&_user=111111111&_pii=S0022404916300846&_rdoc=1&_issn=00224049&md5=d47e27c30971d1948fb178d009b290f1" title="Click to view the MathML source">V=Ker(π)span><span class="mathContainer hidden"><span class="mathCode">span>span>span>. All algebra structures on E such that <span id="mmlsi1" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300846&_mathId=si1.gif&_user=111111111&_pii=S0022404916300846&_rdoc=1&_issn=00224049&md5=950723cc4f86a0c5bae3c7dda614bdbd" title="Click to view the MathML source">π:E→Aspan><span class="mathContainer hidden"><span class="mathCode">span>span>span> becomes an algebra map are described and classified by an explicitly constructed global cohomological type object <span id="mmlsi3" class="mathmlsrc">source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300846&_mathId=si3.gif&_user=111111111&_pii=S0022404916300846&_rdoc=1&_issn=00224049&md5=20a827da2224eb798b41a806189d7cd3">![]()
ss="imgLazyJSB inlineImage" height="18" width="83" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022404916300846-si3.gif">script>![]()
style="vertical-align:bottom" width="83" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022404916300846-si3.gif"> script><span class="mathContainer hidden"><span class="mathCode">span>span>span>. Any such algebra is isomorphic to a Hochschild product <span id="mmlsi111" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300846&_mathId=si111.gif&_user=111111111&_pii=S0022404916300846&_rdoc=1&_issn=00224049&md5=774239def1031e83c3d5db38a448633b" title="Click to view the MathML source">A⋆Vspan><span class="mathContainer hidden"><span class="mathCode">span>span>span>, an algebra introduced as a generalization of a classical construction. We prove that <span id="mmlsi3" class="mathmlsrc">source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300846&_mathId=si3.gif&_user=111111111&_pii=S0022404916300846&_rdoc=1&_issn=00224049&md5=20a827da2224eb798b41a806189d7cd3">![]()
ss="imgLazyJSB inlineImage" height="18" width="83" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022404916300846-si3.gif">script>![]()
style="vertical-align:bottom" width="83" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022404916300846-si3.gif"> script><span class="mathContainer hidden"><span class="mathCode">span>span>span> is the coproduct of all non-abelian cohomologies <span id="mmlsi6" class="mathmlsrc">source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300846&_mathId=si6.gif&_user=111111111&_pii=S0022404916300846&_rdoc=1&_issn=00224049&md5=b6c82aff1a59034cbdc13b19bb6365a5">![]()
ss="imgLazyJSB inlineImage" height="18" width="93" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022404916300846-si6.gif">script>![]()
style="vertical-align:bottom" width="93" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022404916300846-si6.gif"> script><span class="mathContainer hidden"><span class="mathCode">span>span>span>. The key object <span id="mmlsi278" class="mathmlsrc">source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300846&_mathId=si278.gif&_user=111111111&_pii=S0022404916300846&_rdoc=1&_issn=00224049&md5=78a3455e1d9c08abb0b87f86e08af52b">![]()
ss="imgLazyJSB inlineImage" height="18" width="79" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022404916300846-si278.gif">script>![]()
style="vertical-align:bottom" width="79" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022404916300846-si278.gif"> script><span class="mathContainer hidden"><span class="mathCode">span>span>span> responsible for the classification of all co-flag algebras is computed. All Hochschild products <span id="mmlsi298" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300846&_mathId=si298.gif&_user=111111111&_pii=S0022404916300846&_rdoc=1&_issn=00224049&md5=a4c10dd5a52ed4e465515fa3e7119ea9" title="Click to view the MathML source">A⋆kspan><span class="mathContainer hidden"><span class="mathCode">span>span>span> are also classified and the automorphism groups <span id="mmlsi9" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300846&_mathId=si9.gif&_user=111111111&_pii=S0022404916300846&_rdoc=1&_issn=00224049&md5=3fe379aff930185d34df3096e4bbb487" title="Click to view the MathML source">Aut<sub>Algsub>(A⋆k)span><span class="mathContainer hidden"><span class="mathCode">span>span>span> are fully determined as subgroups of a semidirect product <span id="mmlsi10" class="mathmlsrc">source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022404916300846&_mathId=si10.gif&_user=111111111&_pii=S0022404916300846&_rdoc=1&_issn=00224049&md5=9e2d7085e46daead01f25886808957cc">![]()
ss="imgLazyJSB inlineImage" height="20" width="165" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022404916300846-si10.gif">script>![]()
style="vertical-align:bottom" width="165" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022404916300846-si10.gif"> script><span class="mathContainer hidden"><span class="mathCode">span>span>span> 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.