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This is a concise introduction to Fomin-Zelevinskys cluster algebras and their links with the representation theory of quivers in the acyclic case. We review the definition of cluster algebras (geometric, without coefficients), construct the cluster category and present the bijection between cluster variables and rigid indecomposable objects of the cluster category.
We prove that the quantum cluster algebra structure of a unipotent quantum coordinate ring $A_q(mathfrak{n}(w))$, associated with a symmetric Kac-Moody algebra and its Weyl group element $w$, admits a monoidal categorification via the representations
Let $Q$ be a finite acyclic valued quiver. We give the high-dimensional cluster multiplication formulas in the quantum cluster algebra of $Q$ with arbitrary coefficients, by applying certain quotients of derived Hall subalgebras of $Q$.
In cite{CK2005} and cite{Hubery2005}, the authors proved the cluster multiplication theorems for finite type and affine type. We generalize their results and prove the cluster multiplication theorem for arbitrary type by using the properties of 2--Ca
We introduce and investigate new invariants on the pair of modules $M$ and $N$ over quantum affine algebras $U_q(mathfrak{g})$ by analyzing their associated R-matrices. From new invariants, we provide a criterion for a monoidal category of finite-dim
For a finite-dimensional gentle algebra, it is already known that the functorially finite torsion classes of its category of finite-dimensional modules can be classified using a combinatorial interpretation, called maximal non-crossing sets of string