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张量范畴（影印版）

样章
作者：
Pavel Etingof， Shlomo Gelaki， Dmitri Nikshych，Victor Ostrik
定价：
169.00元
版面字数：
580.00千字
开本：
16开
装帧形式：
精装
版次：
1
最新版次
印刷时间：
2026-01-05
ISBN：
978-7-04-066130-9
物料号：
66130-00
出版时间：
2026-03-27
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
分析
暂无
  • 前辅文
  • Preface
  • Chapter 1. Abelian categories
    • 1.1. Categorical prerequisites and notation
    • 1.2. Additive categories
    • 1.3. Definition of abelian category
    • 1.4. Exact sequences
    • 1.5. Length of objects and the Jordan-H¨older theorem
    • 1.6. Projective and injective objects
    • 1.7. Higher Ext groups and group cohomology
    • 1.8. Locally finite (artinian) and finite abelian categories
    • 1.9. Coalgebras
    • 1.10. The Coend construction
    • 1.11. Deligne’s tensor product of locally finite abelian categories
    • 1.12. The finite dual of an algebra
    • 1.13. Pointed coalgebras and the coradical filtration
    • 1.14. Bibliographical notes
  • Chapter 2. Monoidal categories
    • 2.1. Definition of a monoidal category
    • 2.2. Basic properties of unit objects
    • 2.3. First examples of monoidal categories
    • 2.4. Monoidal functors and their morphisms
    • 2.5. Examples of monoidal functors
    • 2.6. Monoidal functors between categories of graded vector spaces
    • 2.7. Group actions on categories and equivariantization
    • 2.8. The Mac Lane strictness theorem
    • 2.9. The coherence theorem
    • 2.10. Rigid monoidal categories
    • 2.11. Invertible objects and Gr-categories
    • 2.12. 2-categories
    • 2.13. Bibliographical notes
  • Chapter 3. Z+-rings
    • 3.1. Definition of a Z+-ring
    • 3.2. The Frobenius-Perron theorem
    • 3.3. The Frobenius-Perron dimensions
    • 3.4. Z+-modules
    • 3.5. Graded based rings
    • 3.6. The adjoint based subring and universal grading
    • 3.7. Complexified Z+-rings and *-algebras
    • 3.8. Weak based rings
    • 3.9. Bibliographical notes
  • Chapter 4. Tensor categories
    • 4.1. Tensor and multitensor categories
    • 4.2. Exactness of the tensor product
    • 4.3. Semisimplicity of the unit object
    • 4.4. Absence of self-extensions of the unit object
    • 4.5. Grothendieck ring and Frobenius-Perron dimension
    • 4.6. Deligne’s tensor product of tensor categories
    • 4.7. Quantum traces, pivotal and spherical categories
    • 4.8. Semisimple multitensor categories
    • 4.9. Grothendieck rings of semisimple tensor categories
    • 4.10. Categorification of based rings
    • 4.11. Tensor subcategories
    • 4.12. Chevalley property of tensor categories
    • 4.13. Groupoids
    • 4.14. The adjoint subcategory and universal grading
    • 4.15. Equivariantization of tensor categories
    • 4.16. Multitensor categories over arbitrary fields
    • 4.17. Bibliographical notes
  • Chapter 5. Representation categories of Hopf algebras
    • 5.1. Fiber functors
    • 5.2. Bialgebras
    • 5.3. Hopf algebras
    • 5.4. Reconstruction theory in the infinite setting
    • 5.5. More examples of Hopf algebras
    • 5.6. The quantum group Uq(sl2)
    • 5.7. The quantum group Uq(g)
    • 5.8. Representations of quantum groups and quantum function algebras
    • 5.9. Absence of primitive elements
    • 5.10. The Cartier-Gabriel-Kostant theorem
    • 5.11. Pointed tensor categories and Hopf algebras
    • 5.12. Quasi-bialgebras
    • 5.13. Quasi-bialgebras with an antipode and quasi-Hopf algebras
    • 5.14. Twists for bialgebras and Hopf algebras
    • 5.15. Bibliographical notes
    • 5.16. Other results
  • Chapter 6. Finite tensor categories
    • 6.1. Properties of projective objects
    • 6.2. Categorical freeness
    • 6.3. Injective and surjective tensor functors
    • 6.4. The distinguished invertible object
    • 6.5. Integrals in quasi-Hopf algebras and unimodular categories
    • 6.6. Degeneracy of the Cartan matrix
    • 6.7. Bibliographical notes
  • Chapter 7. Module categories
    • 7.1. The definition of a module category
    • 7.2. Module functors
    • 7.3. Module categories over multitensor categories
    • 7.4. Examples of module categories
    • 7.5. Exact module categories over finite tensor categories
    • 7.6. First properties of exact module categories
    • 7.7. Module categories and Z+-modules
    • 7.8. Algebras in multitensor categories
    • 7.9. Internal Homs in module categories
    • 7.10. Characterization of module categories in terms of algebras
    • 7.11. Categories of module functors
    • 7.12. Dual tensor categories and categorical Morita equivalence
    • 7.13. The center construction
    • 7.14. The quantum double construction for Hopf algebras
    • 7.15. Yetter-Drinfeld modules
    • 7.16. Invariants of categorical Morita equivalence
    • 7.17. Duality for tensor functors and Lagrange’s Theorem
    • 7.18. Hopf bimodules and the Fundamental Theorem
    • 7.19. Radford’s isomorphism for the fourth dual
    • 7.20. The canonical Frobenius algebra of a unimodular category
    • 7.21. Categorical dimension of a multifusion category
    • 7.22. Davydov-Yetter cohomology and deformations of tensor categories
    • 7.23. Weak Hopf algebras
    • 7.24. Bibliographical notes
    • 7.25. Other results
  • Chapter 8. Braided categories
    • 8.1. Definition of a braided category
    • 8.2. First examples of braided categories and functors
    • 8.3. Quasitriangular Hopf algebras
    • 8.4. Pre-metric groups and pointed braided fusion categories
    • 8.5. The center as a braided category
    • 8.6. Factorizable braided tensor categories
    • 8.7. Module categories over braided tensor categories
    • 8.8. Commutative algebras and central functors
    • 8.9. The Drinfeld morphism
    • 8.10. Ribbon monoidal categories
    • 8.11. Ribbon Hopf algebras
    • 8.12. Characterization of Morita equivalence
    • 8.13. The S-matrix of a pre-modular category
    • 8.14. Modular categories
    • 8.15. Gauss sums and the central charge
    • 8.16. Representation of the modular group
    • 8.17. Modular data
    • 8.18. The Anderson-Moore-Vafa property and Verlinde categories
    • 8.19. A non-spherical generalization of the S-matrix
    • 8.20. Centralizers and non-degeneracy
    • 8.21. Dimensions of centralizers
    • 8.22. Projective centralizers
    • 8.23. De-equivariantization
    • 8.24. Braided G-crossed categories
    • 8.25. Braided Hopf algebras, Nichols algebras, pointed Hopf algebras
    • 8.26. Bibliographical notes
    • 8.27. Other results
  • Chapter 9. Fusion categories
    • 9.1. Ocneanu rigidity (absence of deformations)
    • 9.2. Induction to the center
    • 9.3. Duality for fusion categories
    • 9.4. Pseudo-unitary fusion categories
    • 9.5. Canonical spherical structure
    • 9.6. Integral and weakly integral fusion categories
    • 9.7. Group-theoretical fusion categories
    • 9.8. Weakly group-theoretical fusion categories
    • 9.9. Symmetric and Tannakian fusion categories
    • 9.10. Existence of a fiber functor
    • 9.11. Deligne’s theorem for infinite categories
    • 9.12. The Deligne categories Rep(St), Rep(GLt), Rep(Ot), Rep(Sp2t)
    • 9.13. Recognizing group-theoretical fusion categories
    • 9.14. Fusion categories of prime power dimension
    • 9.15. Burnside’s theorem for fusion categories
    • 9.16. Lifting theory
    • 9.17. Bibliographical notes
    • 9.18. Other results
  • Bibliography
  • Index

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