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.
