弱Hopf代数表示范畴的精确序列
Exact sequences of representation categories of weak Hopf algebras
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中文总结 AI 辅助
本文研究弱Hopf代数表示范畴的精确序列，给出代数条件并发展余核理论，证明余核为Hopf代数，并举例说明构造。
中文摘要 AI 辅助
我们研究了任意域上弱Hopf代数表示范畴的精确序列。给定一个序列$A\overset{k}{\to} B\overset{\pi}{\to} H$，其中$A$和$B$是弱Hopf代数，$H$是一个Hopf代数，我们发展了关于$k$和$\pi$的可验证的代数条件，在这些条件下，存在一个Bruguières和Natale（2011）意义下的张量范畴的精确序列$\mathrm{Rep}(H)\to\mathrm{Rep}(B)\to\mathrm{Rep}(A)$。在此过程中，我们发展了对于结合代数之间的映射$\pi:B\to H$的标量限制函子的推广，该推广满足一个依赖于相对可分子代数$B_r\subseteq B$的弱化乘法约束，以及弱Hopf代数的核与余核理论。我们特别发现，弱Hopf代数同态$k:A\to B$的余核总是存在，它是一个Hopf代数，并且当$A$是连通的时，余核映射是满射的。我们最后通过研究由群胚、拟三角弱Hopf代数的形式ribbon扩张以及一个弱作用于弱Hopf代数的Hopf代数的余循环交叉积构造的此类精确序列的例子来结束本文。
英文摘要
We study exact sequences of representation categories of weak Hopf algebras over an arbitrary field. Given a sequence $A\overset{k}{\to} B\oversetπ{\to} H$, where $A$ and $B$ are weak Hopf algebras and $H$ is a Hopf algebra, we develop verifiable algebraic conditions on $k$ and $π$ under which there is an exact sequence $\mathrm{Rep}(H)\to\mathrm{Rep}(B)\to\mathrm{Rep}(A)$ of tensor categories in the sense of Bruguières and Natale (2011). Along the way, we develop a generalization of the restriction of scalars functor for maps $π:B\to H$ between associative algebras satisfying a weakened multiplicativity constraint depending on a relatively separable subalgebra $B_r\subseteq B$, as well as a theory of kernels and cokernels for weak Hopf algebras. We, in particular, find that the cokernel of a weak Hopf algebra homomorphism $k:A\to B$ always exists, is a Hopf algebra, and the cokernel map is surjective when $A$ is connected. We conclude by studying examples of such exact sequences built from groupoids, formal ribbon extensions of quasitriangular weak Hopf algebras, and cocycled crossed products of a Hopf algebra acting weakly on a weak Hopf algebra.
发表机构
- University of California, Santa Barbara（加州大学圣塔芭芭拉分校）
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