The Manin Hopf algebra of a Koszul Artin-Schelter regular algebra is quasi-hereditary

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• 作者：Theo Raedschelders^a ; ¹ ; ^{traedsch@vub.ac.be" class="auth_mail" title="E-mail the corresponding author} ; Michel Van den Bergh^b ; ² ; ^{michel.vandenbergh@uhasselt.be" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 16S10 ; 16S37 ; 16S38 ; 16T05 ; 16T15 ; 20G42
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：10 January 2017
• 年：2017
• 卷：305
• 期：Complete
• 页码：601-660
• 全文大小：815 K

文摘

For any Koszul Artin–Schelter regular algebra A  , we consider the universal Hopf algebra ce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815302358&_mathId=si1.gif&_user=111111111&_pii=S0001870815302358&_rdoc=1&_issn=00018708&md5=18b12949cea9071659c7d0d9bfa4b0b1">ce" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870815302358-si1.gif">$\underset{\_}{\mathrm{aut}}(A)$ coacting on A  , introduced by Manin. To study the representations (i.e. finite dimensional comodules) of this Hopf algebra, we use the Tannaka–Krein formalism. Specifically, we construct an explicit combinatorial rigid monoidal category ce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815302358&_mathId=si108.gif&_user=111111111&_pii=S0001870815302358&_rdoc=1&_issn=00018708&md5=6930d00c572e42b936b1118f353ab672" title="Click to view the MathML source">U$\mathcal{U}$, equipped with a functor M   to finite dimensional vector spaces such that ce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815302358&_mathId=si1013.gif&_user=111111111&_pii=S0001870815302358&_rdoc=1&_issn=00018708&md5=a753538bfe52b215d37be954dbd10aa2">ce" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870815302358-si1013.gif">$\underset{\_}{\mathrm{aut}}(A)={\mathrm{coend}}_{\mathcal{U}}(M)$. Using this pair ce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815302358&_mathId=si4.gif&_user=111111111&_pii=S0001870815302358&_rdoc=1&_issn=00018708&md5=4448c54fcfaa060482b4a4738bf7fe9f" title="Click to view the MathML source">(U,M)$(\mathcal{U},M)$ we show that ce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815302358&_mathId=si1.gif&_user=111111111&_pii=S0001870815302358&_rdoc=1&_issn=00018708&md5=18b12949cea9071659c7d0d9bfa4b0b1">ce" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870815302358-si1.gif">$\underset{\_}{\mathrm{aut}}(A)$ is quasi-hereditary as a coalgebra and in addition is derived equivalent to the representation category of ce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870815302358&_mathId=si108.gif&_user=111111111&_pii=S0001870815302358&_rdoc=1&_issn=00018708&md5=6930d00c572e42b936b1118f353ab672" title="Click to view the MathML source">U$\mathcal{U}$.