Abstract
F-polynomials and g-vectors were defined by Fomin and Zelevinsky to give a formula which expresses cluster variables in a cluster algebra in terms of the initial cluster data. A quantum cluster algebra is a certain noncommutative deformation of a cluster algebra. In this thesis, we define and prove the existence of analogous quantum F -polynomials for quantum cluster algebras. We compute quantum F-polynomials and g-vectors for a certain class of cluster variables, which includes all cluster variables in type A quantum cluster algebras. Finally, we give formulas for F-polynomials and quantum F-polynomials in classical types when the initial exchange matrix is acyclic.
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