Monoidal categorification and quantum affine algebras

We introduce and investigate new invariants of pairs of modules $M$ and $N$ over quantum affine algebras $U_{q}^{\prime }(\mathfrak{g})$ by analyzing their associated $R$ -matrices. Using these new invariants, we provide a criterion for a monoidal category of finite-dimensional integrable $U_{q}^{\prime }(\mathfrak{g})$ -modules to become a monoidal categorification of a cluster algebra.