通过Ringel–Hall李代数构造GIM和椭圆李代数
GIM and Elliptic Lie algebras via Ringel--Hall Lie algebras
浏览论文内容
中文总结 AI 辅助
该研究通过构造带对合的无环赋值箭图及其轨道范畴，利用Ringel–Hall李代数建立GIM代数与椭圆李代数的相关同构与同态，证实了实根空间上的单射性。
中文摘要 AI 辅助
对于任意可对称化广义相交矩阵（Generalized Intersection Matrix，GIM）$C$，我们构造一个带有对合$\theta$的无环赋值箭图$(Q,\boldsymbol{d})$。设$\boldsymbol{\textit{D}}$为$(Q,\boldsymbol{d})$的有限维表示的有界导出范畴，$\boldsymbol{\textit{\u03A3}}$为$\boldsymbol{\textit{D}}$的平移函子。我们证明轨道范畴$\boldsymbol{\textit{D}}/(\theta\boldsymbol{\u2218}\boldsymbol{\u03A3}}$具有典型的三角化结构且是2-周期的。将Peng–Xiao的构造应用于该轨道范畴，我们证明GIM代数$\boldsymbol{\textit{gim}}(C)$同构于与$\boldsymbol{\textit{D}}/(\theta\boldsymbol{\u2218}\boldsymbol{\u03A3}}$相关的整Ringel–Hall李代数。作为上述方法的进一步应用，我们研究$D_4^{(1,1)}$、$E_6^{(1,1)}$、$E_7^{(1,1)}$和$E_8^{(1,1)}$型椭圆李代数。对于每个椭圆Dynkin图，我们通过取附属于GIM矩阵$C$的无环箭图$Q$的适当商来定义有限维代数$A$。从所得的2-周期三角化范畴出发，我们构造对应的Ringel–Hall李代数，并建立从每个椭圆李代数到其整Ringel–Hall对应物的满李代数同态。该映射被猜测是单射的，且其在实根空间上的单射性已得到证实。
英文摘要
For any symmetrizable generalized intersection matrix (GIM) $C$, we construct an acyclic valued quiver $(Q,\mathbf{d})$ endowed with an involution $θ$. Let $\mathcal{D}$ be the bounded derived category of finite-dimensional representations of $(Q,\mathbf{d})$, and let $Σ$ stand for the suspension functor of $\mathcal{D}$. We show that the orbit category $\mathcal{D}/(θ\circΣ)$ carries a canonical triangulated structure and is $2$-periodic. Applying Peng--Xiao's construction to this orbit category, we prove that the GIM algebra $\operatorname{gim}(C)$ is isomorphic to the integral Ringel--Hall Lie algebra associated with $\mathcal{D}/(θ\circΣ)$. As a further application of the above machinery, we investigate elliptic Lie algebras of types $D_4^{(1,1)}$, $E_6^{(1,1)}$, $E_7^{(1,1)}$ and $E_8^{(1,1)}$. For each elliptic Dynkin diagram, we define a finite-dimensional algebra $A$ by taking an appropriate quotient of the acyclic quiver $Q$ attached to the GIM matrix $C$. From the resulting $2$-periodic triangulated categories, we build the corresponding Ringel--Hall Lie algebras, and establish a surjective Lie algebra homomorphism from each elliptic Lie algebra to its integral Ringel--Hall counterpart. This map is conjectured to be injective, and its injectivity on real root spaces is confirmed.
