Related papers: Monoidal categorification of cluster algebras II

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…

We summarize the definition of the Weyl groupoid using supercategory approach in order to investigate quantum superalgebras at roots of unity. We show how the structure of a Hopf superalgebra on a quantum superalgebra is determined by the…

We summarize the definition of the Weyl groupoid using supercategory approach in order to investigate quantum superalgebras at roots of unity. We show how the structure of a Hopf superalgebra on a quantum superalgebra is determined by the…

The quantum co-ordinate algebra $A_{q}(\mathfrak{g})$ associated to a Kac-Moody Lie algebra $\mathfrak{g}$ forms a Hopf algebra whose comodules are precisely the $U_{q}(\mathfrak{g})$ modules in the BGG category…

This paper consists of three parts: (I) To develop general theory of a (large) class of central simple finite dimensional algebras and answering some natural questions about them (that in general situation it is not even clear how to…

In this paper we categorify the q-Schur algebra S(n,d) as a quotient of Khovanov and Lauda's diagrammatic 2-category U(sln). We also show that our 2-category contains Soergel's monoidal category of bimodules of type A, which categorifies…

Let $Q$ be the affine quiver of type $\widetilde{A}_{2n-1,1}$ and $\mathcal{A}_{q}(Q)$ be the quantum cluster algebra associated to the valued quiver $(Q,(2,2,\dots,2))$. We prove some cluster multiplication formulas, and deduce that the…

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…

Let g be a symmetrisable Kac-Moody algebra and V an integrable g-module in category O. We show that the monodromy of the (normally ordered) rational Casimir connection on V can be made equivariant with respect to the Weyl group W of g, and…

Let $A$ be the path algebra of a finite acyclic quiver $Q$ over a finite field. We realize the quantum cluster algebra with principal coefficients associated to $Q$ as a sub-quotient of a certain Hall algebra involving the category of…

We construct a family of right coideal subalgebras of quantum groups, which have the property that all irreducible representations are one-dimensional, and which are maximal with this property. The obvious examples for this are the standard…

Let $k$ be a field and $A$ a finite-dimensional $k$-algebra of global dimension $\leq 2$. We construct a triangulated category $\Cc_A$ associated to $A$ which, if $A$ is hereditary, is triangle equivalent to the cluster category of $A$.…

Let $W$ be the Weyl group corresponding to a finite dimensional simple Lie algebra $\mathfrak{g}$ of rank $\ell$ and let $m>1$ be an integer. In [I21], by applying cluster mutations, a $W$-action on $\mathcal{Y}_m$ was constructed. Here…

We consider skew-symmetrizable (upper) cluster algebras with a compatible Poisson structure, called $\mathsf{\Lambda}$-(upper) cluster algebras. For any two good elements (e.g., cluster monomials) in a $\mathsf{\Lambda}$-upper cluster…

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…

Various coordinate rings of varieties appearing in the theory of Poisson Lie groups and Poisson homogeneous spaces belong to the large, axiomatically defined class of symmetric Poisson nilpotent algebras, e.g. coordinate rings of Schubert…

We define the notion of braided Coxeter category, which is informally a tensor category carrying compatible, commuting actions of a generalised braid group B_W and Artin's braid groups B_n on the tensor powers of its objects. The data which…

We study the set S of labelled seeds of a cluster algebra of rank n inside a field F as a homogeneous space for the group M_n of (globally defined) mutations and relabellings. Regular equivalence relations on S are associated to subgroups W…

We give a complete classification of right coideal subalgebras that contain all group-like elements for the quantum group $U_q^+(\frak{so}_{2n+1}),$ provided that $q$ is not a root of 1. If $q$ has a finite multiplicative order $t>4,$ this…

We determine the set of dominant $\ell$--weights in the Weyl (or standard) modules for quantum affine $A_n$. We then prove that the space of homomorphisms between standard modules is at most one-dimensional and give a necessary and…