Related papers: A generalized quantum cluster algebra of Kronecker…
Lots of research focuses on the combinatorics behind various bases of cluster algebras. This paper studies the natural basis of a type A cluster algebra, which consists of all cluster monomials. We introduce a new kind of combinatorial…
A generalization of the Heisenberg algebra has been recently constructed. This generalized algebra has a characteristic function which depends on one of its generators. When this function is linear, $qJ_0+s$, it is possible to construct a…
We prove a number of results having to do with equipping type-I $\mathrm{C}^*$-algebras with compact quantum group structures, the two main ones being that such a compact quantum group is necessarily co-amenable, and that if the…
We establish ring isomorphisms between quantum Grothendieck rings of certain remarkable monoidal categories of finite-dimensional representations of quantum affine algebras of types $A_{2n-1}^{(1)}$ and $B_n^{(1)}$. Our proof relies in part…
A general framework for obtaining certain types of contracted and centrally extended algebras is presented. The whole process relies on the existence of quadratic algebras, which appear in the context of boundary integrable models.
We construct, using the quantum dilogarithm, a series of *-representations of quantized cluster varieties. This includes a construction of infinite dimensional unitary projective representations of their discrete symmetry groups - the…
We construct the quantized enveloping algebra of any simple Lie algebra of type ADE as the quotient of a Grothendieck ring arising from certain cyclic quiver varieties. In particular, the dual canonical basis of a one-half quantum group…
We study the lower bound algebras generated by the generalized projective cluster variables of acyclic generalized cluster algebras of geometric types. We prove that this lower bound algebra coincides with the corresponding generalized…
We discuss a generalization of Kummer construction which, on the base of an integral representation of a finite group and local resolution of its quotient, produces a higher dimensional variety with trivial canonical class. As an…
We give the Ringel-Hall algebra construction of the positive half of quantum Borcherds-Bozec algebras as the generic composition algebras of quivers with loops.
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…
It has been proved in \cite{LS} that cluster variables in cluster algebras of every skew-symmetric cluster algebra are positive. We prove that any regular generalized cluster variable of an affine quiver is positive. As a corollary, we…
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 calculate the cluster modular groups of affine and doubly extended typecluster algebras in a uniform way by introducing a new family of quivers. We use this uniformdescription to construct a natural finite quotient of the cluster complex…
We generalize the construction of the bracelet and bangle bases defined by Musiker, Schiffler and Williams, and the band basis defined by D. Thurston to cluster algebras arising from orbifolds. We prove that the bracelet bases are positive,…
In \cite{JKS} we gave an (additive) categorification of Grassmannian cluster algebras, using the category $\CM(A)$ of Cohen-Macaulay modules for a certain Gorenstein order $A$. In this paper, using a cluster tilting object in the same…
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…
We suggest a definition of cluster polylogarithms on an arbitrary cluster variety and classify them in type $A$. We find functional equations for multiple polylogarithms which generalize equations discovered by Abel, Kummer, and Goncharov…
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…
In this paper we give a direct proof of the positivity conjecture for adapted quantum cluster variables. Moreover, our process allows one to explicitly compute formulas for all adapted cluster monomials and certain ordered products of…