جبر دیاگرامی (پژوهش‌های ریاضی و مونوگراف‌ها، جلد ۲۶۴)

This book is an introduction to techniques and results in diagrammatic algebra. It starts with abstract tensors and their categorifications, presents diagrammatic methods for studying Frobenius and Hopf algebras, and discusses their relations with topological quantum field theory and knot theory. The text is replete with figures, diagrams, and suggestive typography that allows the reader a glimpse into many higher dimensional processes. The penultimate chapter summarizes the previous material by demonstrating how to braid 3- and 4- dimensional manifolds into 5- and 6-dimensional spaces. The book is accessible to post-qualifier graduate students, and will also be of interest to algebraists, topologists and algebraic topologists who would like to incorporate diagrammatic techniques into their research." Provided by publisher Chapter 1. Introduction Chapter 2. Elements 1. Sets, relations, and functions 2. Diagrammatics of linear algebra 3. Algebras 4. Simplfying, clarifying, and abstracting 5. General categorical principles Chapter 3. Planar trivalent diagrams 1. Small categories 2. Trivalent graphs Chapter 4. The multi-category FA 1. Composition of double arrows 2. Reversible arrows and additional double arrows Chapter 5. Triple arrows for FA 1. Algebraic identities as double arrows 2. The category of double arrows 3. Critical aspects of weakly invertible 1-arrows 4. Coalgebra axioms as triple arrows Chapter 6. Surfaces in 3-space 1. Guide to terminology 2. Objects, 1-arrows, and double arrows 3. Triple arrows in S 4. Quadruple arrows in the multi-category S 5. Functorial equivalent multi-categories Chapter 7. Beyond surfaces 1. Different objects and arrows 2. Weak inverses revisited 3. Higher order arrows in FA 4. Restricting the collections of arrows Chapter 8. Parentheses and so forth 1. The Temperley-Lieb algebra 2. Other Catalan-like things 3. Higher associativities 4. Higher dimensional foams Chapter 9. Knots in space 1. Oriented knots and higher categories 2. Reidemeister moves 3. The fundamental group and related invariants 4. The Jones polynomial 5. The braid group 6. More algebraic structures 7. Trivalent graphs Chapter 10. Foams and surfaces in 4-space 1. Knotted surfaces 2. Foams in 4-space 3. Shalgebras and qualgebras 4. Homology 5. More abstract tensors 6. Conclusion Chapter 11. Higher dimensional braids 1. Geometric braids 2. Glyphographic description of surface braids 3. Surface braids 4. Charts in 3- and 4-dimensions 5. Conclusion Chapter 12. Globular multi-categories 1. Arrows and cells 2. Group presentations 3. Conclusion Bibliography Index Offers an introduction to techniques and results in diagrammatic algebra. The book starts with abstract tensors and their categorifications, presents diagrammatic methods for studying Frobenius and Hopf algebras, and discusses their relations with topological quantum field theory and knot theory.