Related papers: On the acyclic quantum cluster algebras with princ…

Let $Q$ be an affine quiver of type $A_2^{(1)}$. We explicitly construct the cluster multiplication formulas for the quantum cluster algebra of $Q$ with principal coefficients. As applications, we obtain: (1)\ an exact expression for every…

Inspired by a previous work of Nakajima, we consider perverse sheaves over acyclic graded quiver varieties and study the Fourier-Sato-Deligne transform from a representation theoretic point of view. We obtain deformed monoidal…

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 establish a quantum cluster algebra structure on the quantum Grothendieck ring of a certain monoidal subcategory of the category of finite-dimensional representations of a simply-laced quantum affine algebra. Moreover, the…

Let $\mathbb{X}_{\boldsymbol{p},\boldsymbol{\lambda}}$ be a weighted projective line. We define the quantum cluster algebra of $\mathbb{X}_{\boldsymbol{p},\boldsymbol{\lambda}}$ and realize its specialized version as the subquotient of the…

Let $Q$ be a finite quiver without oriented cycles and $\mathcal A(Q)$ be the coefficient-free cluster algebra with initial seed $(Q,\textbf u)$. Using the Caldero-Chapoton map, we introduce and investigate a family of generic variables in…

The cluster multiplication formulas for a generalized quantum cluster algebra of Kronecker type are explicitly given. Furthermore, a positive bar-invariant $\mathbb{Z}[q^{\pm\frac{1}{2}}]$-basis of this algebra is constructed.

Let $Q$ be an acyclic quiver. We introduce the notion of generic variables for the coefficient-free acyclic cluster algebra $\mathcal A(Q)$. We prove that the set $\mathcal G(Q)$ of generic variables contains naturally the set $\mathcal…

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…

We construct cluster algebras the variables and coefficients of which satisfy the discrete mKdV equation, the discrete Toda equation and other integrable bilinear equations, several of which lead to q-discrete Painlev\'e equations. These…

We show that the quantum coordinate ring of the unipotent subgroup N(w) of a symmetric Kac-Moody group G associated with a Weyl group element w has the structure of a quantum cluster algebra. This quantum cluster structure arises naturally…

We develop an elementary formula for certain non-trivial elements of upper cluster algebras. These elements have positive coefficients. We show that when the cluster algebra is acyclic these elements form a basis. Using this formula, we…

We introduce a framework for $\mathbb{Z}$-gradings on cluster algebras (and their quantum analogues) that are compatible with mutation. To do this, one chooses the degrees of the (quantum) cluster variables in an initial seed subject to a…

The cluster algebra of any acyclic quiver can be realized as the coordinate ring of a subvariety of a Kac-Moody group -- the quiver is an orientation of its Dynkin diagram, defining a Coxeter element and thereby a double Bruhat cell. We use…

Geiss-Leclerc-Schroer defined the cluster algebra structure on the coordinate ring $C[N(w)]$ of the unipotent subgroup, associated with a Weyl group element $w$ and they proved cluster monomials are contained in Lusztig's dual semicanonical…

These notes are mainly based on arXiv:2003.13674 and a series of talks given in the workshop CARTEA. For any symmetrizable Kac-Moody algebra $\mathfrak{g}$ and any Weyl group element $w$, the corresponding quantum unipotent subgroup…

For an algebraically closed field $K$, we investigate a class of noncommutative $K$-algebras called connected quantized Weyl algebras. Such an algebra has a PBW basis for a set of generators $\{x_1,\dots,x_n\}$ such that each pair satisfies…

The article gives a ring theoretic perspective on cluster algebras. Gei{\ss}-Leclerc-Schr\"oer prove that all cluster variables in a cluster algebra are irreducible elements. Furthermore, they provide two necessary conditions for a cluster…

For a rooted cluster algebra $\mathcal{A}(Q)$ over a valued quiver $Q$, a \emph{symmetric cluster variable} is any cluster variable belonging to a cluster associated with a quiver $\sigma (Q)$, for some permutation $\sigma$. The subalgebra…

We introduce a quantisation of the Coxeter-Conway frieze patterns and prove that they realise quantum cluster variables in quantum cluster algebras associated with linearly oriented Dynkin quivers of type A. As an application, we obtain the…