论文标题
具有相等内核属性的尺寸向量
Dimension vectors with the equal kernels property
论文作者
论文摘要
令$ r \ in \ mathbb n $，$γ_r$为$ r $ arrows $γ_1，\ ldots，γ_r\ colon 1 \ to 2 $ to 2 $ to和$δ\inΔ_+（γ_r）$的广义KRONECKER QUIVER。我们说$δ$具有相等的内核属性，如果对于所有$α\ in k^r \ setMinus \ {0 \} $以及每个不可兼容表示$ m $，带有尺寸vector $ \ usevector $ \ usevection {dim} m =δ M_1 \至M_2 $是注入的。我们表明，$δ$在且仅当$ q_ {γ_r}（δ） +δ_2-Δ_1\ geq 1 $时，其中$ q_ {γ_r} \ colon \ colon \ colon \ mathbb z^2 \ to \ mathb z，（x，y） $γ_R$。
Let $r \in \mathbb N$, $Γ_r$ be the generalized Kronecker quiver with $r$ arrows $γ_1,\ldots,γ_r \colon 1 \to 2$ and $δ\in Δ_+(Γ_r)$ be a positive root of $Γ_r$. We say that $δ$ has the equal kernels property if for all $α\in k^r \setminus \{0\}$ and every indecomposable representation $M$ with dimension vector $\underline{dim} M = δ$ the $k$-linear map $M^α:= \sum^r_{i=1} α_i M(γ_i) \colon M_1 \to M_2$ is injective. We show that $δ$ has the equal kernels property if and only if $q_{Γ_r}(δ) + δ_2 - δ_1 \geq 1$, where $q_{Γ_r} \colon \mathbb Z^2 \to \mathbb Z, (x,y) \mapsto x^2 + y^2 - rxy$ denotes the Tits quadratic form of $Γ_r$.
