Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 3 / 3

The extension group ${\mathrm{Ext}}_Q^1(V,W)$ consists of equivalence classes of short exact sequences $0\to W\to E\to V\to 0$, with the zero class represented by a split sequence and addition given by the Baer sum. The extension complex of quiver representations givesThe printed map has $\gamma (u{)}_{\rho }=u_j{f}_{\rho }-g_{\rho }{u}_i$, so its kernel is ${\mathrm{Hom}}_Q(V,W)$ and its cokernel is ${\mathrm{Ext}}_Q^1(V,W)$. Reversing the overall differential sign changes neither identification.

For dimension vectors $\mathbf{v},\mathbf{w}$, the Ringel form isThe first expression makes its dependence only on the dimension vectors explicit.

For the one-loop representation $V=k$ with loop scalar $\lambda$, $\gamma (u)=u\lambda -\lambda u=0$ on $k$. Both cochain spaces have dimension one, so $\dim {\mathrm{Ext}}_Q^1(V,V)=1$, for every $\lambda$. Concretely, a self-extension has loop matrix $\left(\begin{array}{cc}\lambda & c\\ 0& \lambda \end{array}\right)$, with $c$ the extension parameter.

The four-subspace quiver has four one-dimensional sources and a two-dimensional sink. Its Tits form of a quiver at this dimension vector is $q=4\cdot 1^2+2^2-4(1\cdot 2)=0$.

An endomorphism comprises source scalars $a_1,\dots ,a_4$ and a sink matrix $T$. The first two columns $e_1,e_2$ force $T=\mathrm{diag}(a_1,a_2)$. The third column $e_1+e_2$ then forces $a_1=a_2=a_3$, making $T$ scalar. Since the fourth column $(\lambda ,1{)}^T$ is always nonzero, its scalar is also the same, for every $\lambda$.

Thus ${\mathrm{End}}_Q(V)=k$, so the representation is a brick module. The Ringel form givesThis conclusion also covers $\lambda =0,1$, where the fourth line repeats one of the earlier lines; the first three lines already force scalar endomorphisms.