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We introduce and investigate new invariants of pairs of modules
over quantum affine algebras
by analyzing their associated
-matrices. Using these new invariants, we provide a criterion for a monoidal category of finite-dimensional integrable
-modules to become a monoidal categorification of a cluster algebra.
We prove that, for simple modules
over a quantum affine algebra, their tensor product
has a simple head and a simple socle if
is simple. A similar result is proved for the convolution product of simple modules over quiver Hecke algebras.
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