Quantization of canonical bases and the quantum symplectic double
  We describe a natural $q$-deformation of Fock and Goncharov's canonical basis for the algebra of regular functions on a cluster variety associated to a quiver of type $A$. We then describe an extension of this construction involving a cluster variety called the symplectic double. 
Dylan G. L. Allegretti
math.QA hep-th math.GT