发表机构
School of Mathematical Sciences, Fudan University; School of Mathematics, Shanghai University of Finance and Economics（复旦大学数学科学学院; 上海财经大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文建立无特征框架，将单位根处量子群的同调性质归结为小量子群，并证明大量子群为Artin-Schelter Gorenstein及给出Calabi-Yau判据。
AI 中文摘要
我们发展了一个一般的无特征框架，用于研究一大类模有限Hopf代数的同调性质。该框架使得将许多单位根处量子群及其多参数形变的同调性质的研究简化为相应小量子群的研究成为可能。对于任何仿射Hopf代数$H$，若其具有一个大的中心Hopf子代数$C$，我们证明其左同调积分空间（在Lu-Wu-Zhang的意义下）作为双模与恒等纤维代数的左积分空间同构，而恒等纤维代数是有限维Hopf代数。因此，$H$是$C$的对称Frobenius扩张当且仅当相应的恒等纤维代数是幺模的且$H$的对极的平方是内自同构，从而为$H$的Calabi-Yau性质提供了有效判据。对于一大类单位根处量子群，可以选择合适的大中心Hopf子代数使得恒等纤维代数就是相应的小量子群。因此，这些大量子群的某些同调性质由其相应的小量子群的同调性质所控制。假设基域是代数闭的，我们证明$H$是幺模的当且仅当对于$C$的某个（等价地，每个）极大理想$\mathfrak{m}$，纤维代数在$\mathfrak{m}$处的有限维表示范畴在Yadav的意义下作为恒等纤维代数的有限维表示构成的有限张量范畴上的模范畴是幺模的。作为应用，我们证明所有Andruskiewitsch-Angiono-Yakimov大量子群都是仿射诺特幺模Artin-Schelter Gorenstein Hopf代数。我们还给出了这些大量子群成为Calabi-Yau的充要条件。
英文摘要
We develop a general characteristic-free framework for studying the homological properties of a broad class of module-finite Hopf algebras. This framework makes it possible to reduce the study of the homological properties of many quantum groups at roots of unity and their multiparameter deformations to the study of the corresponding small quantum groups. For any affine Hopf algebra $H$ admitting a large central Hopf subalgebra $C$, we prove that its left homological integral space, in the sense of Lu-Wu-Zhang, is isomorphic as a bimodule to the left integral space of the identity fiber algebra, which is a finite-dimensional Hopf algebra. Consequently, $H$ is a symmetric Frobenius extension of $C$ if and only if the corresponding identity fiber algebra is unimodular and the square of the antipode of $H$ is inner, thus providing an effective criterion for the Calabi-Yau property of $H$. For a broad class of quantum groups at roots of unity, an appropriate large central Hopf subalgebra can be chosen such that the identity fiber algebra is the corresponding small quantum group. Therefore, some homological properties of these big quantum groups are governed by those of their corresponding small quantum groups. Assuming that the base field is algebraically closed, we prove that $H$ is unimodular if and only if, for some (equivalently, every) maximal ideal $\mathfrak{m}$ of $C$, the category of finite-dimensional representations of the fiber algebra at $\mathfrak{m}$ is unimodular in the sense of Yadav as a module category over the finite tensor category of finite-dimensional representations of the identity fiber algebra. As an application, we prove that all Andruskiewitsch-Angiono-Yakimov large quantum groups are affine noetherian unimodular Artin-Schelter Gorenstein Hopf algebras. We also give a necessary and sufficient condition for these large quantum groups to be Calabi-Yau.
Comments46 pages
