Symplectic quivers (Q1910984)

Let \(\Gamma\) be a finite connected graph with an orientation \(\Lambda\). Let \(\Gamma_0\) be the set of vertices and let \(\Gamma_1\) be the set of edges. For every edge \(l\in\Gamma_1\) let \(\alpha(l)\) denote the initial vertex and let \(\beta(l)\) denote the final vertex. To each vertex \(\alpha\in\Gamma_0\) we assign a finite-dimensional linear space with a skew-symmetric 2-form \((V_\alpha,\omega_\alpha)\). To each edge \(l\in\Gamma_1\) we assign a linear mapping \(f_l:V_{\alpha(l)}\to V_{\beta(l)}\) such that \(f^*_l\omega_{\beta(l)}=\omega_{\alpha l}\). Let \((V,\omega,f)\) be the set of spaces \(V_\alpha\), forms \(\omega_\alpha\), and mappings \(f_l\). For the oriented graph \((\Gamma,\Lambda)\), let us define a category \({\mathcal S}(\Gamma,\Lambda)\) as follows. The objects of this category are the sets \((V,\omega,f)\) and the morphisms \(\varphi:(V,\omega,f)\to(W,\gamma,g)\) are the sets of linear mappings \(\varphi_\alpha:V_\alpha\to W_\alpha\), \(\alpha\in\Gamma_0\), such that \(\varphi^*_\alpha\gamma_\alpha=\omega_\alpha\) and for each edge \(l\in\Gamma_1\) the obvious diagram is commutative. Objects of the category \({\mathcal S}(\Gamma,\Lambda)\) are called symplectic quivers. In the category \({\mathcal S}(\Gamma,\Lambda)\) we can consider the subcategory \({\mathcal S}_0(\Gamma,\Lambda)\) consisting of the quivers for which all the mappings \(f_l\) are epimorphisms. The category \({\mathcal S}_0(\Gamma,\Lambda)\) contains indecomposable objects, and any quiver from \({\mathcal S}_0(\Gamma,\Lambda)\) can be decomposed into a direct sum of these objects. We prove that the category \({\mathcal S}_0(\Gamma,\Lambda)\) contains only a finite set of indecomposable quivers if and only if the graph \(\Gamma\) coincides with one of the following Dynkin diagrams: \(A_n\), \(D_n\), \(E_6\), \(E_7\), and \(E_8\), and the orientation on these diagrams can be chosen arbitrarily.