Foundations of Quantum Group Theory

This is a graduate-level text that systematically develops the foundations of the subject. Quantum groups (i.e. Hopf algebras) are treated as mathematical objects in their own right; basic properties and theorems are proven in detail from this standpoint, including the results underlying key applications. After formal definitions and basic theory, the book goes on to cover such topics as quantum enveloping algebras, matrix quantum groups, combinatorics, cross products of various kinds, the quantum double, the semiclassical theory of Poisson-Lie groups, the representation theory, braided groups and applications to q-deformed physics. The explicit proofs and a great many worked examples and exercises will allow readers to quickly pick up the techniques needed for working in this exciting new field. Now in paperback, this is a graduate level text for theoretical physicists and mathematicians which systematically lays out the foundations for the subject of Quantum Groups in a clear and accessible way. The topic is developed in a logical manner with quantum groups (Hopf Algebras) treated as mathematical objects in their own right. After formal definitions and basic theory, the book goes on to cover such topics as quantum enveloping algebras, matrix quantum groups, combinatorics, cross products of various kinds, the quantum double, the semiclassical theory of Poisson-Lie groups, the representation theory, braided groups and applications to q-deformed physics. Explicit proofs and many examples will allow the reader quickly to pick up the techniques needed for working in this exciting new field. 1. Definition Of Hopf Algebras -- 2. Quasitriangular Hopf Algebras -- 3. Quantum Enveloping Algebras -- 4. Matrix Quantum Groups -- 5. Quantum Random Walks And Combinatorics -- 6. Bicrossproduct Hopf Algebras -- 7. Quantum Double And Double Cross Products -- 8. Lie Bialgebras And Poisson Brackets -- 9. Representation Theory -- 10. Braided Groups And Q-deformation. Shahn Majid. Includes Bibliographical References And Index.