In equation (4.16) of https://arxiv.org/abs/1506.06601, a discretization of the (classical) affine Kac-Moody algebra is presented:
$$ \frac{1}{\gamma}\left\{J_{m}^{1}, J^2_{n}\right\}=J_{m}^{1} J_{n}^{2} \overline{\mathfrak{r}} \delta_{m n}-\overline{\mathfrak{r}}^{t} {J}_{m}^{1} J_{n}^{2} \delta_{m n}-J_{m}^{1} \overline{\mathfrak{r}} J_{n}^{2} \delta_{m, n+1}+J_{n}^{2} \overline{\mathfrak{r}}^{t} J_{m}^{1} \delta_{m+1, n},\tag{4.16} $$
where $A^1=A\otimes 1$ and $B^2=1\otimes B$. Here $\overline{\mathfrak{r}}$ is a solution of the classical Yang-Baxter equation that satisfies
$$ \overline{\mathfrak{r}}+\overline{\mathfrak{r}}^{t}=2 \mathfrak{C},\tag{4.15} $$
where $\mathfrak{C}$ is the tensor Casimir of the associated Lie algebra, i.e., $\sum_aT_a\otimes T_a$.
The claim is that by expanding $$ J_{n} \sim 1+\frac{2 \pi}{k} \Delta \mathscr{J}_{+}(x),\tag{4.14} $$
we obtain
$$ \left\{\mathscr{J}_{+}^{1}(x), \mathscr{\mathscr { J }}_{+}^{2}(y)\right\}=\frac{1}{2}\left[\mathfrak{C}, \mathscr{J}_{+}^{1}(x)-\mathscr{\mathscr { J }}_{+}^2(x)\right] \delta(x-y) \pm \frac{k}{2 \pi} \mathfrak{C} \delta^{\prime}(x-y). $$
My question is, how does one show this? An important step seems to be the identification between the Kronecker and Dirac deltas as explained in Where does the delta of zero $\delta(0)$ come from? , $$ \frac{1}{\Delta}\delta_{mn}\rightarrow \delta(x-y). $$
But it seems that we also need to discretize the derivative of the Dirac delta, $\delta'(x-y)$, and it ought to be expressible in terms of $\Delta^2\delta_{m+1,n}$. How does one do this?
