组合Hopf代数与量子化问题研究

组合Hopf代数源于对称函数理论，发展至今与表示论范畴化、数学物理中拓扑量子场论等国际热点领域密切交融。由于组合Hopf代数的乘法与余乘结构具有非负整性，寻求恰当的表示论赋予其范畴化解释，是这类数学对象的核心研究问题之一。探寻代数结构量子化是现代数学研究的新使命，涉及表示论方法、形变同调论、张量范畴理论等方面。本项目探讨组合Hopf代数与量子群理论之间内在联系，关注（辫子）Hopf代数结构建立，研究如下几个问题：.（1）统一实现根树Hopf代数模型的辫子化；.（2）范畴化与量子化Connes-Kreimer根树Hopf代数；.（3）研究微分代数形变理论及其在量子泛包络代数上的反映；.（4）从表示论和辫子张量范畴角度研究量子超群“Majid猜想的超版本”与量子微分算子实现。.项目成果将为（辫子）Hopf代数理论以及辫子张量范畴研究提供丰富内涵，也为单位根处小量子超群的模表示论提供新研究路径。

The research of combinatorial Hopf algebras originates from the theory of symmetric functions, and has developed to become a direction, interacting with many active fields such as categorification by representation theory and quantum field theory in mathematical physics. Based on the integral positivity of multiplication and comultiplication of a combinatorial Hopf algebra, finding proper representation theory to give it categorical interpretation, is one of the central problems around this subject. On the other hand, quantization of a given algebraic structure is a new challenge in modern mathematics, involving several aspects like representation theory, deformation theory, and the theory of tensor categories, etc. This project studies the intrinsic relationship between combinatorial Hopf algebras and quantum group theory, concerns about the construction of (braided) Hopf algebras, and attacks the following questions:.(1) To give the braided analogue of Hopf algebra models of rooted trees under a uniform machinery;.(2) To categorify and quantize the Connes-Kreimer Hopf algebra of planar rooted trees;.(3) To study deformation theory of differential algebras, and its application to the quantum enveloping algebras; .(4) To solve the super-version of Majid's conjecture for quantum supergroups and their quantum differential operators' realizations from the perspective of representation theory and braided tensor categories..The achievement of this project will provide rich contents for the study of (braided) Hopf algebras or even braided tensor categories, and also a new strategy for the modular representation theory of small quantum supergroups at roots of unity.