The Classical and Quantum 6j-symbols. (MN-43) - Softcover

Carter, J. Scott; Flath, Daniel E.; Saito, Masahico

Softcover

ISBN 10: 0691027307 ISBN 13: 9780691027302

Publisher: Princeton University Press, 1995

Synopsis

Addressing physicists and mathematicians alike, this book discusses the finite dimensional representation theory of sl(2), both classical and quantum. Covering representations of U(sl(2)), quantum sl(2), the quantum trace and color representations, and the Turaev-Viro invariant, this work is useful to graduate students and professionals.

The classic subject of representations of U(sl(2)) is equivalent to the physicists' theory of quantum angular momentum. This material is developed in an elementary way using spin-networks and the Temperley-Lieb algebra to organize computations that have posed difficulties in earlier treatments of the subject. The emphasis is on the 6j-symbols and the identities among them, especially the Biedenharn-Elliott and orthogonality identities. The chapter on the quantum group Ub-3.0 qb0(sl(2)) develops the representation theory in strict analogy with the classical case, wherein the authors interpret the Kauffman bracket and the associated quantum spin-networks algebraically. The authors then explore instances where the quantum parameter q is a root of unity, which calls for a representation theory of a decidedly different flavor. The theory in this case is developed, modulo the trace zero representations, in order to arrive at a finite theory suitable for topological applications. The Turaev-Viro invariant for 3-manifolds is defined combinatorially using the theory developed in the preceding chapters. Since the background from the classical, quantum, and quantum root of unity cases has been explained thoroughly, the definition of this invariant is completely contained and justified within the text.

About this title

From the Back Cover:

This book discusses the representation theory of classical and quantum U ( sl(2) ) with an eye towards topological applications of the latter. We use the Temperley-Lieb algebra and the quantum spin-networks to organize the computations. We define the 6j-symbols in the classical, quantum, and quantum-root-of-unity cases, and use these computations to define the Turaev-viro invariants of closed 3-dimensional manifolds.

About the Author:

J. Scott Carter is Associate Professor and Daniel E. Flath is Associate Professor, both in the Department of Mathematics at the University of South Alabama. Masahico Saito is Assistant Professor of Mathematics at the University of South Florida.

About this edition

• Publisher: Princeton University Press
• Publication date: 1995
• ISBN 10: 0691027307
• ISBN 13: 9780691027302
• Binding: Paperback
• Number of pages: 168