Also appears in collection : Exposés de recherche

The theory of cluster algebras has proved useful in proving theorems about the characters of graded tensor products or Demazure modules, via the $Q$-system. Upon quantization, the algebra associated with this system is shown to be related to a quantum affine algebra. Graded characters are related to a polynomial representation of the quantum cluster variables. This immediately suggests a further deformation to the spherical DAHA, quantum toroidal algebras and elliptic Hall algebras.