发表机构
Mathematical Institute of Charles University; Alma Mater Studiorum - Università di Bologna（查理大学数学研究所; 博洛尼亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究在辫子幺正范畴发展微分演算理论，结合双积构造协变smash积微分演算，引入量子G-结构并以实例说明，为量子几何提供新理论框架。
AI 中文摘要
我们在辫子幺正范畴中发展了一阶微分演算理论，并对辫子Hopf代数上的辫子协变及双协变微分演算进行分类。我们证明，在特定条件下，双协变微分演算可转化为辫子双协变微分演算。对于Radford-Majid双积，我们将Hopf代数上的双协变微分演算与对应辫子Hopf代数上的辫子双协变微分演算相结合，得到协变 smash 积微分演算。相关的Maurer-Cartan形式可分解为量子主丛的结构Hopf代数的Maurer-Cartan形式与基上辫子Hopf代数的Maurer-Cartan形式的直和。从几何角度看，该构造实现了给定Hopf代数的量子仿射扩张，且我们证明所得量子主丛配备了由量子Maurer-Cartan形式诱导的标架消解。基于此对应，我们引入并发展量子G-结构的概念，证明量子G-结构是约化上的量子标架消解。该理论通过Sweedler Hopf代数高维类似物的转化实例及辫子量子平面（视为O_q(GL₂)的Yetter-Drinfeld模）得到说明。
英文摘要
We develop a theory of first order differential calculi in braided monoidal categories and classify braided covariant and bicovariant calculi on braided Hopf algebras. We show that, under certain conditions, bicovariant calculi can be transmuted to braided bicovariant calculi. For Radford--Majid biproducts, we combine bicovariant calculi on a Hopf algebra and braided bicovariant calculi on the corresponding braided Hopf algebra to covariant smash product calculi. The associated Maurer--Cartan form is shown to decompose into a direct sum of the Maurer--Cartan forms of the structure Hopf algebra of the quantum principal bundle and the braided Hopf algebra on the base. Geometrically, this construction realises the quantum affine extension of a given Hopf algebra, and we prove that the resulting quantum principal bundle is equipped with a frame resolution induced by the quantum Maurer--Cartan form. Building on this correspondence, we introduce and develop the notion of quantum $\textrm{G}$-structure, proving that quantum $\textrm{G}$-structures are quantum frame resolutions on the reduction. The theory is illustrated by examples based on transmutations of higher analogues of Sweedler's Hopf algebra and on the braided quantum plane, seen as a Yetter--Drinfeld module of $O_q(\mathrm{GL}_2)$.
Comments49 pages, comments are welcome
