We first study a new family of graded quiver varieties together with a new $t$-deformation of the associated Grothendieck rings. This provides the geometric foundations for a joint paper by Yoshiyuki Kimura and the author. 
We further generalize the result of that paper to any acyclic quantum cluster algebra with arbitrary nondegenerate coefficients. In particular, we obtain the generic basis, the dual PBW basis, and the dual canonical basis. The method consists in a correction technique, which works for general quantum cluster algebras.

$t$-Analog of $q$-Characters, Bases of Quantum Cluster Algebras, and a Correction Technique

t-Analog of q-Characters, Bases of Quantum Cluster Algebras, and a Correction Technique

Multiplicative properties of a quantum Caldero–Chapoton map associated to valued quivers

In Berenstein and Rupel (2015), the authors defined algebra homomorphisms from the dual Ringel-Hall algebra of certain hereditary abelian category $\mathcal {A}$ to an appropriate q-polynomial algebra. In the case that $\mathcal {A}$ is the representation category of an acyclic quiver, we give an alternative proof by using the cluster multiplication formulas in (Ding and Xu, Sci. China Math. 55(10) 2045–2066, 2012). Moreover, if the underlying graph of Q associated with $\mathcal {A}$ is bipartite and the matrix B associated to the quiver Q is of full rank, we show that the image of the algebra homomorphism is in the corresponding quantum cluster algebra.

On Homomorphisms from Ringel-Hall Algebras to Quantum Cluster Algebras

Let $Q$ be a finite acyclic valued quiver. We give the cluster multiplication formulas in the quantum cluster algebra of $Q$ with arbitrary coefficients, by applying certain quotients of derived Hall subalgebras of $Q$. These formulas can be viewed as the quantum version of the cluster multiplication theorem in the classical cluster algebra proved by Caldero-Keller for finite type, Hubery for affine type and Xiao-Xu for acyclic quivers.

The cluster multiplication theorem for acyclic quantum cluster algebras.

We extend the definition of a quantum analogue of the Caldero-Chapoton map defined in \cite{rupel}. When $Q$ is a quiver of finite type, we prove that the algebra $\mathcal{AH}_{|k|}(Q)$ generated by all cluster characters (see Definition \ref{def}) is exactly the quantum cluster algebra $\mathcal{EH}_{|k|}(Q)$.

On quantum cluster algebras of finite type

We extend the definition of a quantum analogue of the Caldero-Chapoton map defined by D. Rupel. When Q is a quiver of finite type, we prove that the algebra (Q) generated by all cluster characters is exactly the quantum cluster algebra (Q).

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On quantum cluster algebras of finite type

On quantum cluster algebras of finite type