Quantum Cluster Algebras and Fusion Products

Q-systems are recursion relations satisfied by the characters of the restrictions of special finite-dimensional modules of quantum affine algebras. They can also be viewed as mutations in certain cluster algebras, which have a natural quantum deformation. In this paper, we explain the relation in the simply laced case between the resulting quantum Q-systems and the graded tensor product of Feigin and Loktev. We prove the graded version of the M=N identities, and write expressions for these as noncommuting evaluated multiresidues of suitable products of solutions of the quantum Q-system. This leads to a simple reformulation of Feigin and Loktev’s fusion coefficients as matrix elements in a representation of the quantum Q-system algebra.