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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 of symmetric Khovanov-Lauda- Rouquier algebras. In order to achieve this goal, we give a formulation of monoidal categorifications of quantum cluster algebras and provide a criterion for a monoidal category of finite-dimensional graded $R$-modules to become a monoidal categorification, where $R$ is a symmetric Khovanov-Lauda-Rouquier algebra. Roughly speaking, this criterion asserts that a quantum monoidal seed can be mutated successively in all the directions, once the first-step mutations are possible. Then, we show the existence of a quantum monoidal seed of $A_q(mathfrak{n}(w))$ which admits the first-step mutations in all the directions. As a consequence, we prove the conjecture that any cluster monomial is a member of the upper global basis up to a power of $q^{1/2}$. In the course of our investigation, we also give a proof of a conjecture of Leclerc on the product of upper global basis elements.
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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-dimensional integrable $U_q(mathfrak{g})$-modules to become a monoidal categorification of a cluster algebra.
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 introduce a new family of real simple modules over the quantum affine algebras, called the affine determinantial modules, which contains the Kirillov-Reshetikhin (KR)-modules as a special subfamily, and then prove T-systems among them which generalize the T-systems among KR-modules and unipotent quantum minors in the quantum unipotent coordinate algebras simultaneously. We develop new combinatorial tools: admissible chains of i-boxes which produce commuting families of affine determinantial modules, and box moves which describe the T-system in a combinatorial way. Using these results, we prove that various module categories over the quantum affine algebras provide monoidal categorifications of cluster algebras. As special cases, Hernandez-Leclerc categories provide monoidal categorifications of the cluster algebras for an arbitrary quantum affine algebra.
Let $U_q(mathfrak{g})$ be a quantum affine algebra of untwisted affine $ADE$ type, and $mathcal{C}_{mathfrak{g}}^0$ the Hernandez-Leclerc category of finite-dimensional $U_q(mathfrak{g})$-modules. For a suitable infinite sequence $widehat{w}_0= cdots s_{i_{-1}}s_{i_0}s_{i_1} cdots$ of simple reflections, we introduce subcategories $mathcal{C}_{mathfrak{g}}^{[a,b]}$ of $mathcal{C}_{mathfrak{g}}^0$ for all $a le b in mathbb{Z}sqcup{ pm infty }$. Associated with a certain chain $mathfrak{C}$ of intervals in $[a,b]$, we construct a real simple commuting family $M(mathfrak{C})$ in $mathcal{C}_{mathfrak{g}}^{[a,b]}$, which consists of Kirillov-Reshetikhin modules. The category $mathcal{C}_{mathfrak{g}}^{[a,b]}$ provides a monoidal categorification of the cluster algebra $K(mathcal{C}_{mathfrak{g}}^{[a,b]})$, whose set of initial cluster variables is $[M(mathfrak{C})]$. In particular, this result gives an affirmative answer to the monoidal categorification conjecture on $mathcal{C}_{mathfrak{g}}^-$ by Hernandez-Leclerc since it is $mathcal{C}_{mathfrak{g}}^{[-infty,0]}$, and is also applicable to $mathcal{C}_{mathfrak{g}}^0$ since it is $mathcal{C}_{mathfrak{g}}^{[-infty,infty]}$.
In 2009, Keller and Yang categorified quiver mutation by interpreting it in terms of equivalences between derived categories. Their approach was based on Ginzburgs Calabi-Yau algebras and on Derksen-Weyman-Zelevinskys mutation of quivers with potential. Recently, Matthew Pressland has generalized mutation of quivers with potential to that of ice quivers with potential. In this paper, we show that his rule yields derived equivalences between the associated relative Ginzburg algebras, which are special cases of Yeungs deformed relative Calabi-Yau completions arising in the theory of relative Calabi-Yau structures due to Toen and Brav-Dyckerhoff. We illustrate our results on examples arising in the work of Baur-King-Marsh on dimer models and cluster categories of Grassmannians. We also give a categorification of mutation at frozen vertices as it appears in recent work of Fraser-Sherman-Bennett on positroid cluster structures.
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