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

### Customer Reviews:

#### Great book, great service

By **S. L. Starr** - July 12, 2010
#### Good introduction to topological quantum field theory

By **Dr. Lee D. Carlson** - June 3, 2001

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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 6*j*-symbols and the identities among them, especially the Biedenharn-Elliott and orthogonality identities. The chapter on the quantum group *Uq(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.

This book arrived from the book seller extremely fast. I ordered it at the end of a trip to visit a colleague, and it was in my mailbox by the time I got home. The trip took a few days. I'm a fan of Amazon's used book service. The book is very nice. It is a nice supplement to Kaufman and Lins's book about Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds. It is also nice to read in conjunction with Frenkel and Khovanov's Duke paper on the graphical representation of the dual canonical basis. I particularly like this book's pictures, such as the movie/cartoon type picture of moving 2-complexes to explain the 6j symbols as relating to the skeleton of the tetrahedron. I also like the graphical proof of the Elliott-Biedenharn identity. I have a few questions about all these things that do not appear in the books. For example, in Theorem 3.12(1) they state an identity for a sum with an alternating sign. Is this also related to Rota's q-analogue of the principal of inclusion,... read more

This book is an excellent introduction to the concepts and techniques used to define invariants of closed 3-dimensional manifolds using the representation theory of U(sl(2)). Starting with the well-known results in the finite-dimensional irreducible representations of SL(2) via the Clebsch-Gordan theory, one can decompose the tensor product of these representations in two ways. The two decompositions can be compared using recoupling theory, with the coefficients being the ubiquitous 6j-symbols, so familiar to physicists in the theory of angular momentum. The orthogonality and Elliott-Biedenharn identities of the 6j-symbols have a geometric interpretation as the union of two tetrahedra. The quantum analog of these results for sl(2) leads to the Turaev-Vivo invariants of 3-manifolds, with the Elliott-Biedenharn identity corresponding to an Alexander move on a triangulation of a 3-manifold and the orthogonality condition corresponding to a Matveev move on the dual 2-skeleton of a... read more

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