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

The Classical and Quantum 6j-symbols. (MN-43)
J. Scott Carter,Daniel E. Flath,Masahico Saito | 1995-12-11 00:00:00 | Princeton University Press | 168 | Mathematics
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 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.
Reviews
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, exclusion? It would be nice if there was a book that reviewed this topic from all the different perspectives: quantum spin systems, quantum invariants for geometry, quantum algebra, combinatorics and quantum information theory. But this book is a classic and I'm very glad I bought it.
Reviews
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 triangulation.

The book could thus be considered an introduction to the theory of "quantum topology". The authors employ many diagrams to illustrate the beautiful connections between topology and algebra using the reprensentations of U(sl(2)) and the "quantized" version where the representation spaces are homogeneous polynomials in two variables that commute modulo a parameter. These constructions are generalizations of the ones that are employed in studying exactly solved models in statistical mechanics using the Yang-Baxter equation. This theory is now called quantum groups, even though strictly speaking, the objects dealt with are more general than groups and the adjective "quantum" means only a lack of commutation up to a parameter (usually called q). Very interesting is the way in which braid groups appear as realizations of quantum representation spaces. Quotient representations have to be considered since in general the representations of the braid group are not semi-simple.

For a representation of Uq(sl(2)) the authors define trace, called the "quantum trace", in this representation which gives the required invariants. These invariants however are not finer than other 3-manifold invariants unfortunately. The authors do show to what extent two 3-manifolds with the same Turaev-Viro invariants are similar, and show the equivalence between the Turaev-Viro and Kauffman-Lins invariants. These invariants are examples of topological quantum field theories, which have grown out of considerations from high energy physics, and which will no doubt continue to be of considerable interest in the future.

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