Related papers: Acyclic quantum cluster algebras via Hall algebras…

In \cite{rupel3},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…

Let $\mathcal{A}_{q}$ be an arbitrary quantum cluster algebra with principal coefficients. We give the fundamental relations between the quantum cluster variables arising from one-step mutations from the initial cluster in…

Let $Q$ be a finite acyclic valued quiver. We define a bialgebra structure and an integration map on the Hall algebra associated to the morphism category of projective representations of $Q$. As an application, we recover the surjective…

Let $Q$ be a finite quiver without oriented cycles and $k$ an algebraically closed field.In this paper we establish a connection between cluster algebras and the representation theory of the path algebra $kQ$, in terms of the spectral…

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…

We describe quantum enveloping algebras of symmetric Kac-Moody Lie algebras via a finite field Hall algebra construction involving Z_2-graded complexes of quiver representations.

In this paper, we focus on a new lower bound quantum cluster algebra which is generated by the initial quantum cluster variables and the quantum projective cluster variables of an acyclic quantum cluster algebra with principal coefficients.…

We provide a homomorphism of algebras from the quantum group $\mathbf{U}^+_v(\mathfrak{g})$ to the corresponding quantum cluster algebra $\mathcal {A}_q$ with principal coefficients. As a by-product, we show that the quantum cluster…

We give a uniform geometric realization for the cluster algebra of an arbitrary finite type with principal coefficients at an arbitrary acyclic seed. This algebra is realized as the coordinate ring of a certain reduced double Bruhat cell in…

We construct quantized versions of generic bases in quantum cluster algebras of finite and affine types. Under the specialization of $q$ and coefficients to 1, these bases are generic bases of finite and affine cluster algebras.

We prove that certain acyclic cluster algebras over the complex numbers are the coordinate rings of holomorphic symplectic manifolds. We also show that the corresponding quantum cluster algebras have no non-trivial prime ideals. This allows…

We define a quantum loop group $\mathbf{U}^+_Q$ associated to an arbitrary quiver $Q=(I,E)$ and maximal set of deformation parameters, with generators indexed by $I \times \mathbb{Z}$ and some explicit quadratic and cubic relations. We…

Integral cluster categories of acyclic quivers have recently been used in the representation-theoretic approach to quantum cluster algebras. We show that over a principal ideal domain, such categories behave much better than one would…

We construct a cluster algebra structure within the quantum cohomology ring of a quiver variety associated with an $A$-type quiver. Specifically, let $Fl:=Fl(N_1,\ldots,N_{n+1})$ denote a partial flag variety of length $n$, and…

Acyclic cluster algebras have an interpretation in terms of tilting objects in a Calabi-Yau category defined by some hereditary algebra. For a given quiver $Q$ it is thus desirable to decide if the cluster algebra defined by $Q$ is acyclic.…

Let $\Uq$ be a quantum group. Regarding a (noncommutative) space with $\Uq$-symmetry as a $\Uq$-module algebra $A$, we may think of equivariant vector bundles on $A$ as projective $A$-modules with compatible $\Uq$-action. We construct an…

We study quantum cluster algebras from marked surfaces without punctures. We express the quantum cluster variables in terms of the canonical submodules. As a byproduct, we obtain the positivity for this class of quantum cluster algebra.

We prove a multiplication theorem for quantum cluster algebras of acyclic quivers. The theorem generalizes the multiplication formula for quantum cluster variables in \cite{fanqin}. We apply the formula to construct some $\mathbb{ZP}$-bases…

We extend our $\imath$Hall algebra construction from acyclic to arbitrary $\imath$quivers, where the $\imath$quiver algebras are infinite-dimensional 1-Gorenstein in general. Then we establish an injective homomorphism from the universal…

We develop the noncommutative geometry (bundles, connections etc.) associated to algebras that factorise into two subalgebras. An example is the factorisation of matrices $M_2(\C)=\C\Z_2\cdot\C\Z_2$. We also further extend the coalgebra…