发表机构
School of Mathematics and Statistics, Heilongjiang University（黑龙江大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过环面Grothendieck环的双参数特化，识别Cartan型多参数量子群，利用谱刚性得到Serre关系，无需薄性假设。
AI 中文摘要
设$Q$为有限单连通型的二分Dynkin箭图，$\mathscr C_Q$为相应的Hernandez--Leclerc范畴。我们研究Fedele--Hernandez环面Grothendieck环的一个双参数特化，并将其一般正部识别为Cartan型多参数量子群。结构输入是恒等式\\[ 2D^{(2)}-D_0=\omega_Q\circ(°_Q,°_Q), \\]其中$D_0,D^{(2)}$是两个有效的环面交换形式，$\omega_Q$是$Q$的反对称Euler形式。在辅助标量扩张之后，它将特化的特征代数实现为在$v=t_0t^{1/2}$处单参数代数的分次双特征扭转。这给出了所需的根分次和PBW分次维数。对于每个源--汇边$r\to s$，这些维数与局部$A$--$Y$交换公式结合，给出$\operatorname{Ad}_{Z_r}$的二次零化子。因此\\[ \operatorname{Spec}_{Z_r}(Z_s)=\{t_0t,t_0^{-1}\}, \\]无需薄性假设或类型特定的特征公式。这两个谱值产生定向多参数Serre关系。取\\[ q_{ii}=t_0^2t,\qquad q_{ij}=t_0^{-1},\qquad q_{ji}=t_0^{-1}t^{-1}\quad(i\to j), \\]且在非边上$q_{ij}=1$，我们得到\\[ U_{\mathbf q}^{+}(\mathfrak g) \simeq \mathscr K_\infty(\mathscr C_Q)\otimes_RK \\]对于$\mathfrak g$为$A_n,D_n,E_6,E_7,E_8$型。
英文摘要
Let $Q$ be a bipartite Dynkin quiver of finite simply-laced type and $\mathscr C_Q$ 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 \[ 2D^{(2)}-D_0=ω_Q\circ(°_Q,°_Q), \] where $D_0,D^{(2)}$ are the two effective toroidal commutation forms and $ω_Q$ 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 $v=t_0t^{1/2}$. 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 \(\operatorname{Ad}_{Z_r}\). Consequently \[ \operatorname{Spec}_{Z_r}(Z_s)=\{t_0t,t_0^{-1}\}, \] without thinness or type-specific character formulas. The two spectral values yield the oriented multiparameter Serre relations. With \[ q_{ii}=t_0^2t,\qquad q_{ij}=t_0^{-1},\qquad q_{ji}=t_0^{-1}t^{-1}\quad(i\to j), \] and \(q_{ij}=1\) on nonedges, we obtain \[ U_{\mathbf q}^{+}(\mathfrak g) \simeq \mathscr K_\infty(\mathscr C_Q)\otimes_RK \] for $\mathfrak g$ of type $A_n,D_n,E_6,E_7,E_8$.
Comments34 pages
