On some embeddings between the cyclotomic quiver Hecke algebras

Abstract:

Let $I$ be a finite index set and let $A=(a_{ij})_{i,j\in I}$ be an arbitrary indecomposable symmetrizable generalized Cartan matrix. Let $Q^+$ be the positive root lattice and $P^+$ the set of dominant weights. For any $\beta \in Q^+$ and $\Lambda \in P^+$, let $\mathscr {R}_{\beta }^{\Lambda }$ be the corresponding cyclotomic quiver Hecke algebra over a field $K$. For each $i\in I$, there is a natural unital algebra homomorphism $\iota _{\beta ,i}$ from $\mathscr {R}_{\beta }^{\Lambda }$ to $e(\beta ,i)\mathscr {R}_{\beta +\alpha _i}^{\Lambda }e(\beta ,i)$. In this paper we show that the homomorphism $\iota _\beta :=\bigoplus _{i\in I}\iota _{\beta ,i}: \mathscr {R}_{\beta }^{\Lambda }\rightarrow \bigoplus _{i\in I}e(\beta ,i)\mathscr {R}_{\beta +\alpha _i}^{\Lambda }e(\beta ,i)$ is always injective unless $\beta =0$ and $\ell (\Lambda )=0$ or $A$ is of finite type and $\beta =\Lambda -w_0\Lambda$, where $w_0$ is the unique longest element in the finite Weyl group associated to the finite Cartan matrix $A$, and $\ell (\Lambda )$ is the level of $\Lambda$.