Gliozzi–Tateo functional dilogarithm identities for simply laced Y-systems

Let $X_r$ and $X'_{r'}$ be simply laced Dynkin diagrams of finite type, with index sets $I$ and $I'$, ranks $r$ and $r'$, and Coxeter numbers $h$ and $h'$. Let $Y_{ii'}(u)$ be a positive real solution of the Y-system $\mathbb{Y}(X_r,X'_{r'})$, indexed by $i\in I$, $i'\in I'$, and $u\in\mathbb{Z}$, and let $L$ denote the Rogers dilogarithm. Gliozzi–Tateo functional dilogarithm identities. One has

$$\frac{6}{\pi^2}\sum_{(i,i')\in I\times I'}\sum_{u=0}^{2(h+h')-1}L\left(\frac{Y_{ii'}(u)}{1+Y_{ii'}(u)}\right)=2hrr',$$

and

$$\frac{6}{\pi^2}\sum_{(i,i')\in I\times I'}\sum_{u=0}^{2(h+h')-1}L\left(\frac{1}{1+Y_{ii'}(u)}\right)=2h'rr'.$$

These identities generalize the constant dilogarithm identities to functional Y-systems. They are presented as a generalization by Gliozzi and Tateo; the periodicity needed for the finite summation range is stated in the source and is now known in full generality.