HOPF代数表示范畴中的MONOIDAL不变量(英文)

本书在Hopf代数表示范畴层面引入一些新的monoidal不变量，这些不变量包括表示范畴的Green环、Casimir数、高阶Frobenius-Schur指标、Grothendieck环、某种类型的多元齐次多项式等。著作主要研究这些不变量在Hopf代数表示理论中所发挥的作用，揭示这些不变量与Hopf代数表示范畴中其它重要研究对象之间的关系，通过具体实例展示这些不变量的具体表现形式等。这些不变量的引入为人们研究Hopf代数表示范畴的结构与分类提供了新的工具，也为人们深入理解与研究monoidal范畴提供了新的视角。本书所展示的一些研究成果对于推动代数表示理论体系的发展与完善，促进Hopf代数、张量范畴等数学分支的交叉与融合具有积极的作用。

Contents
《博士后文库》序言
Preface
Chapter1The Green Rings of Hopf Algebras1
1.1Hopf algebras1
1.2Quantum traces of morphisms4
1.3Bilinear forms on Green rings11
1.4Some ring-theoretical properties19
Chapter2The Green Rings of Spherical Hopf Algebras28
2.1A new bilinear form28
2.2Quotients of Green rings34
2.3Group-like algebra and bi-Frobenius algebra structure38
Chapter3The Stable Green Rings of Hopf Algebras43
3.1Stable Green rings43
3.2Bi-Frobenius algebra structure47
3.3Applications to Radford Hopf algebras49
Chapter4The Casimir Numbers of Green Rings55
4.1The Jacobson semisimplicity of Green rings55
4.2The Green ring of a cyclic group58
4.3The Casimir number of the Green ring of a cyclic group61
Chapter5The Casimir Numbers of Fusion Categories68
5.1Numerical invariants68
5.2Applications to Verlinde modular categories74
5.3Prime factors of Casimir numbers81
5.4Casimir numbers vs. Frobenius-Schur exponents86
Chapter6Higher Frobenius-Schur Indicators in Positive Characteristic90
6.1Characterizations of S2=id90
6.2Some properties of the element u94
6.3Higher Frobenius-Schur indicators97
6.4Monoidal invariants103
Chapter7The Grothendieck Algebras of Smash Product Hopf Algebras106
7.1Smash product Hopf algebras106
7.2Representations of smash product Hopf algebras109
7.3The Grothendieck algebras of smash product Hopf algebras112
Chapter8Invariants from the Sweedler Power Maps on Integrals122
8.1The Sweedler power maps on integrals122
8.2Polynomial invariants129
8.3Examples133
8.4Integrals of the dual of twisted Hopf algebras137
Bibliography144
Index149
编后记151