Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras

Let J be a set of pairs consisting of good Uq′(g)-modules and invertible elements in the base field C(q). The distribution of poles of normalized R-matrices yields Khovanov–Lauda–Rouquier algebras R^J(β) for each β∈ Q⁺. We define a functor F_β from the category of graded R^J(β) -modules to the category of Uq′(g)-modules. The functor F=Q+Fβ sends convolution products of finite-dimensional graded R^J(β) -modules to tensor products of finite-dimensional Uq′(g)-modules. It is exact if R^J is of finite type A, D, E. If V(ϖ₁) is the fundamental representation of Uq′(sl^N) of weight ϖ₁ and J= {(V(ϖ₁) , q^{2 i}) ∣ i∈ Z} , then R^J is the Khovanov–Lauda–Rouquier algebra of type A_∞. The corresponding functor F sends a finite-dimensional graded R^J-module to a module in C_J, where C_J is the category of finite-dimensional integrable Uq′(sl^N)-modules M such that every composition factor of M appears as a composition factor of a tensor product of modules of the form V(ϖ1)q2s(s∈ Z). Focusing on this case, we obtain an abelian rigid graded tensor category T_J by localizing the category of finite-dimensional graded R^J-modules. The functor F factors through T_J. Moreover, the Grothendieck ring of the category C_J is isomorphic to the Grothendieck ring of T_J at q= 1.