This week I have begun to look at Hyperbolic Tetrahedra and their geometry. In the paper ‘6j Symbols for Uq and non-eucledean Tetrahedra‘, Taylor and Woodward relate the semiclassical asymptotics of the 6j symbols for the quantized enveloping algebra Uq(sl2) to the geometry of spherical and hyperbolic tetrahedra.
The quantum 6j symbol is a function of a 6-tuple jab, 1 ≤ a ≤ b ≤ 4. The 6j symbols
for q = 1 were introduced as a tool in atomic spectroscopy
by Racah, and then studied mathematically by Wigner. 6j symbols
for Uq(sl2) were introduced by Kirillov and Reshetikhin, who used them to generalize the Jones knot invariant. Turaev and Viro used them to define three manifold invariants or quantum gravity with a cosmological constant.
I have started doing some preliminary work with sagemath on 3j symbols, 6j symbols, the quantum integer and on the gram matrix.
- Hamiltonian dynamics of a quantum of space (quantumtetrahedron.wordpress.com)