A Semiclassical Limit of Large-Colour State Sums for Semisimple Lie Algebras

The Melvin–Morton–Rozansky theorem relates the large-colour semiclassical limit of the coloured Jones polynomial to the inverse Alexander polynomial. Bar-Natan and Garoufalidis [BNG96] proved the theorem using weight systems and, more generally, established the corresponding formula for arbitrary complex semisimple Lie algebras. We give a direct proof of this semisimple Lie algebra generalization by analysing a state sum. For a complex semisimple Lie algebra $\mathfrak{g}$ and a dominant integral weight $\lambda$, we use the embedding $V_{d\lambda }\hookrightarrow V_{\lambda }^{\otimes d}$ to analyse the state sum. Under the specialization $q=e^{h\mathrm{/}d}$, we show that, as $d\to \mathrm{\infty }$, only the identity terms and those proportional to $E_{\alpha }\otimes F_{\alpha }$ contribute at leading order. It follows that the leading-order contribution decomposes over the positive roots, with each root contribution evaluated as in the ${\mathfrak{s}\mathfrak{l}}_2$ case. Writing $t_{\alpha }=e^{-2(\lambda ,\alpha )h}$, the limit is therefore given by the product of ${\mathrm{\Delta }}_K(t_{\alpha }{)}^{-1}$ over the positive roots $\alpha$.