Spectral Rigidity in Toroidal Grothendieck Rings and Multiparameter Quantum Groups

Let $Q$ be a bipartite Dynkin quiver of finite simply-laced type and ${\mathcal{C}}_Q$ the corresponding Hernandez–Leclerc category. We study a two-parameter specialization of the Fedele–Hernandez toroidal Grothendieck ring and identify its generic positive part with a Cartan-type multiparameter quantum group. The structural input is the identity $$2D^{(2)}-D_0={\omega }_Q\circ ({\mathrm{°}}_Q,{\mathrm{°}}_Q),$$ where $D_0,D^{(2)}$ are the two effective toroidal commutation forms and ${\omega }_Q$ is the antisymmetric Euler form of $Q$. It realizes the specialized character algebra, after an auxiliary scalar extension, as a graded bicharacter twist of the one-parameter algebra at $v=t_0{t}^{1\mathrm{/}2}$. This yields the required root grading and PBW graded dimensions. For every source–sink edge $r\to s$, these dimensions combine with the local $A$–$Y$ commutation formula to give a quadratic annihilator for ${\mathrm{Ad}}_{Z_r}$. Consequently $${\mathrm{Spec}}_{Z_r}(Z_s)=\{t_{0}t,t_0^{-1}\},$$ without thinness or type-specific character formulas. The two spectral values yield the oriented multiparameter Serre relations. With $$q_{ii}=t_0^{2}t,q_{ij}=t_0^{-1},q_{ji}=t_0^{-1}{t}^{-1}(i\to j),$$ and $q_{ij}=1$ on nonedges, we obtain $$U_{\mathbf{q}}^{+}(\mathfrak{g})\simeq {\mathcal{K}}_{\mathrm{\infty }}({\mathcal{C}}_Q){\otimes }_{R}K$$ for $\mathfrak{g}$ of type $A_n,D_n,E_6,E_7,E_8$.