Acyclic Quantum Cluster Algebras

This thesis concerns quantum cluster algebras. For skew-symmetric acyclic quantum cluster algebras, we express the quantum F -polynomials and the quantum cluster monomials in terms of Serre polynomials of quiver Grassmannians of rigid modules. Then we introduce a new family of graded quiver varieties together with a new t-deformation, and generalize Nakajima’s t-analogue of q-characters to these constructions. This allows a deformed monoidal (pseudo-)categorification approach to bases of the quantum cluster algebras. When the initial seed is acyclic, then for any choice of coefficients and quantization, these characters give us a dual PBW basis, a generic basis, and a dual canonical basis with positive structure constants, such that each of the latter two contains all the quantum cluster monomials. As a byproduct, we obtain the positivity conjecture for the quantum cluster algebras which contain acyclic seeds.