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.

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.

This book gets to the heart of science by asking a fundamental question about its essence: what is the true nature of space and time? Both defy modern physics, and scientists find themselves continually searching for answers. This unique volume brings together world leaders in cosmology, particle physics, quantum gravity, mathematics, philosophy and theology, to provide fresh insights into the deep structure of space and time. In an attempt to understand the question, subjects ranging from dark matter to the philosophical and theological implications of spacetime are covered, ensuring that the issue is thoroughly explored. Interesting and thought-provoking answers provide a well-rounded read.