A Quantum Groups Primer

by Shahn Majid

Overview

Here is a self-contained introduction to quantum groups as algebraic objects. Based on the author's lecture notes for the Part III pure mathematics course at Cambridge University, the book is suitable as a primary text for graduate courses in quantum groups or supplementary reading for modern courses in advanced algebra. The material assumes knowledge of basic and linear algebra. Some familiarity with semisimple Lie algebras would also be helpful. The volume is a primer for mathematicians but it will also be useful for mathematical physicists.

Product Details

ISBN-13:	9780521010412
Publisher:	Cambridge University Press
Publication date:	04/04/2002
Series:	London Mathematical Society Lecture Note Series , #292
Pages:	180
Product dimensions:	5.98(w) x 8.98(h) x 0.67(d)

Table of Contents

Preface; 1. Coalgebras, bialgebras and Hopf algebras. Uq(b+); 2. Dual pairing. SLq(2). Actions; 3. Coactions. Quantum plane A2q; 4. Automorphism quantum groups; 5. Quasitriangular structures; 6. Roots of Unity. uq(sl2); 7. q-Binomials; 8. quantum double. Dual-quasitriangular structures; 9. Braided categories; 10 (Co)module categories. Crossed modules; 11. q-Hecke algebras; 12. Rigid objects. Dual representations. Quantum dimension; 13. Knot invariants; 14. Hopf algebras in braided categories; 15. Braided differentiation; 16. Bosonisation. Inhomogeneous quantum groups; 17. Double bosonisation. Diagrammatic construction of uq(sl2); 18. The braided group Uq(n–). Construction of Uq(g); 19. q-Serre relations; 20. R-matrix methods; 21. Group algebra, Hopf algebra factorisations. Bicrossproducts; 22. Lie bialgebras. Lie splittings. Iwasawa decomposition; 23. Poisson geometry. Noncommutative bundles. q-Sphere; 24. Connections. q-Monopole. Nonuniversal differentials; Problems; Bibliography; Index.

Editorial Reviews

'… would serve admirably - as the author suggests - as the basis for a taught graduate course … this book is a clearly written painless read. I can recommend this text as an entry work for those wishing to acquaint themselves with the still popular topic of quantum groups.' A. I. Solomon, Contemporary Physics

'Many intuitive comments and informal remarks, a well chosen set of main examples used systematically in the book and a clear and understandable style make the book very comfortable and useful for students as well as for mathematicians from other fields.' EMS Newsletter

'This monograph is an excellent reference (and often a true 'eye-opener') for researchers working in quantum groups … S. Majid is well-known for his lively and very informative style of writing, and the reviewed book confirms this opinion. Thus the book is very well written, the proofs contain enough details to make them easily readable but still challenging enough to keep students interested … I can full-heartily recommend this work as a basis for a one-term postgraduate course or as an introductory text to all mathematicians who would like to learn quickly the main ideas, techniques and wide range of applications of quantum group theory.' Zentralblatt MATH

From the Publisher

Intended as a graduate-level textbook, this concise primer provides self-contained introduction to quantum groups as algebraic objects. Chapters cover: coalgebras, bialgebras, and Hopf algebras; dual pairing; coactions; automorphism; quasitriangular structures; the roots of unity; q-binomials, quantum double; braided categories; (Co)module categories; q-Hecke algebras; rigid objects; knot invariants; braided differentiation; bosonisation; double bosonisation; the braided group; q-serre relations; R-matrix methods; group, algebra, Hopf algebra factorizations; Lie bialgebra; lie bialgebras; Poisson geometry; and connections. The book assumes a background understanding of basic algebra and linear algebra, as well as some familiarity with semisimple Lie algebras. Majid teaches at the University of London. Annotation c. Book News, Inc., Portland, OR