The relative modular object and Frobenius extensions of finite Hopf algebras

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• 作者：Kenichi Shimizu ^{kshimizu@shibaura-it.ac.jp" class="auth_mail" title="E-mail the corresponding author}
• 关键词：18D10 ; 16T05
• 刊名：Journal of Algebra
• 出版年：2017
• 出版时间：1 February 2017
• 年：2017
• 卷：471
• 期：Complete
• 页码：75-112
• 全文大小：809 K

文摘

For a certain kind of tensor functor a763a" title="Click to view the MathML source">F:C→D$F:\mathcal{C}\to \mathcal{D}$, we define the relative modular object e663a5709915d8f53dca341a" title="Click to view the MathML source">χ_F∈D${\chi }_F\in \mathcal{D}$ as the “difference” between a left adjoint and a right adjoint of F  . Our main result claims that, if a702f7ffcd142642e6a1ef0fa12ae8" title="Click to view the MathML source">C$\mathcal{C}$ and e67a3c2efacd9bfe39" title="Click to view the MathML source">D$\mathcal{D}$ are finite tensor categories, then e605ff15b9e2d33b61fe546" title="Click to view the MathML source">χ_F${\chi }_F$ can be written in terms of a categorical analogue of the modular function on a Hopf algebra. Applying this result to the restriction functor associated to an extension e65b4392e8ddf0f2296" title="Click to view the MathML source">A/B$A/B$ of finite-dimensional Hopf algebras, we recover the result of Fischman, Montgomery and Schneider on the Frobenius type property of e65b4392e8ddf0f2296" title="Click to view the MathML source">A/B$A/B$. We also apply our results to obtain a “braided” version and a “bosonization” version of the result of Fischman et al.