On some embeddings between the cyclotomic quiver hecke algebras

Let I be a finite index set and let A = (aij )i,j∈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 β ∈ Q+ and Λ ∈ P+, let RΛ β be the corresponding cyclotomic quiver Hecke algebra over a field K. For each i ∈ I, there is a natural unital algebra homomorphism ?β,i from RΛ β to e(β, i)RΛ β+αi e(β, i). In this paper we show that the homomorphism ?β := ⊕∈I ?β,i : RΛ β → ⊕∈I e(β, i)RΛ β+αi e(β, i) is always injective unless β = 0 and ∂(Λ) = 0 or A is of finite type and β = Λ? w0Λ, where w0 is the unique longest element in the finite Weyl group associated to the finite Cartan matrix A, and ∂ (Λ) is the level of Λ.