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 $λ$, we use the embedding $V_{dλ}\hookrightarrow V_λ^{\otimes d}$ to analyse the state sum. Under the specialization $q=e^{h/d}$, we show that, as $d\to\infty$, only the identity terms and those proportional to $E_α\otimes F_α$ contribute at leading order. It follows that the leading-order contribution decomposes over the positive roots, with each root contribution evaluated as in the $\mathfrak{sl}_{2}$ case. Writing $t_α=e^{-2(λ,α)h}$, the limit is therefore given by the product of $Δ_{K}(t_α)^{-1}$ over the positive roots $α$.