[Paper Review] Categories over quantum affine algebras and monoidal categorification

Masaki Kashiwara, Myungho Kim+2 more|arXiv (Cornell University)|May 22, 2020

Algebraic structures and combinatorial models9 references4 citations

TL;DR

This paper establishes a monoidal categorification of cluster algebras arising from subcategories of finite-dimensional representations of quantum affine algebras of untwisted affine $ADE$ type. By constructing a real simple commuting family of Kirillov-Reshetikhin modules indexed by admissible chains of intervals in $[a,b]$, it proves that the Grothendieck rings of these subcategories $$\mathscr{C}_{\mathfrak{g}}^{[a,b]}$$ admit cluster algebra structures with initial seeds given by these modules, thereby affirming the monoidal categorification conjecture for $$\mathscr{C}_{\mathfrak{g}}^{-}$ and $$\mathscr{C}_{\mathfrak{g}}^{0}$$.

ABSTRACT

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 $\mathcal{C}_{\mathfrak{g}}^{[-\infty,\infty]}$.

Motivation & Objective

• To provide a monoidal categorification of the Grothendieck rings $K(\mathscr{C}_{\mathfrak{g}}^{[a,b]})$ for subcategories of finite-dimensional $U_q'({\mathfrak{g}})$-modules.
• To establish that these Grothendieck rings carry a cluster algebra structure with initial cluster variables given by a commuting family of Kirillov-Reshetikhin modules.
• To prove the monoidal categorification conjecture for the subcategory $\mathscr{C}_{\mathfrak{g}}^{-} = \mathscr{C}_{\mathfrak{g}}^{[-\infty,0]}$ and extend it to $\mathscr{C}_{\mathfrak{g}}^{0} = \mathscr{C}_{\mathfrak{g}}^{[-\infty,\infty]}$.
• To unify the construction of initial monoidal seeds across all intervals $[a,b]$ via admissible chains of $i$-boxes.

Proposed method

• Introduce subcategories $\mathscr{C}_{\mathfrak{g}}^{[a,b]}$ of the Hernandez-Leclerc category $\mathscr{C}_{\mathfrak{g}}^{0}$ using an infinite sequence of simple reflections $\widehat{w}_0$.
• Define admissible chains $\mathfrak{C}$ of $i$-boxes in intervals $[a,b]$ to parametrize a commuting family $M(\mathfrak{C})$ of Kirillov-Reshetikhin modules.
• Construct a $\Lambda$-admissible monoidal seed $\mathscr{S} = (M(\mathfrak{C}), \widetilde{B})$ in $\mathscr{C}_{\mathfrak{g}}^{[a,b]}$ using $R$-matrix invariants and the criterion from KKOP (2019).
• Show that the exact sequence associated with mutation corresponds to the $T$-system, ensuring mutation equivalence of all $M(\mathfrak{C})$.
• Prove that $K(\mathscr{C}_{\mathfrak{g}}^{[a,b]})$ is isomorphic to a cluster algebra $\mathscr{A}([\mathscr{S}])$ with initial seed $[\mathscr{S}]$, thus achieving monoidal categorification.
• Apply the framework to $\mathscr{C}_{\mathfrak{g}}^{-}$ and $\mathscr{C}_{\mathfrak{g}}^{0}$, recovering known results and extending them uniformly.

Experimental results

Research questions

• RQ1 Does the subcategory $\mathscr{C}_{\mathfrak{g}}^{[a,b]}$ of finite-dimensional $U_q'({\mathfrak{g}})$-modules admit a cluster algebra structure in its Grothendieck ring?
• RQ2 Can the initial cluster variables of $K(\mathscr{C}_{\mathfrak{g}}^{[a,b]})$ be realized as isomorphism classes of real simple modules, specifically Kirillov-Reshetikhin modules?
• RQ3 Is the monoidal categorification conjecture of Hernandez-Leclerc satisfied for $\mathscr{C}_{\mathfrak{g}}^{-}$ and $\mathscr{C}_{\mathfrak{g}}^{0}$ via a uniform construction using admissible chains?
• RQ4 How are the cluster mutations in $K(\mathscr{C}_{\mathfrak{g}}^{[a,b]})$ realized categorically, and do they correspond to the $T$-system relations?
• RQ5 Can the $\Lambda$-admissibility criterion be applied to construct initial monoidal seeds for all intervals $[a,b]$ in a consistent way?

Key findings

• The subcategory $\mathscr{C}_{\mathfrak{g}}^{[a,b]}$ provides a monoidal categorification of the cluster algebra $\mathscr{A}([\mathscr{S}])$, where $[\mathscr{S}]$ is the initial seed with cluster variables given by $M(\mathfrak{C})$ for an admissible chain $\mathfrak{C}$.
• The Grothendieck ring $K(\mathscr{C}_{\mathfrak{g}}^{[a,b]})$ is isomorphic to a cluster algebra with initial seed $[\mathscr{S}]$, confirming the cluster algebra structure.
• The monoidal seed $\mathscr{S}^{-} = (M(\mathfrak{C}^{-}), \widetilde{B}^{-})$ for $\mathscr{C}_{\mathfrak{g}}^{-} = \mathscr{C}_{\mathfrak{g}}^{[-\infty,0]}$ is $\Lambda$-admissible, thus affirming the monoidal categorification conjecture for $\mathscr{C}_{\mathfrak{g}}^{-}$.
• The initial cluster variables of $K(\mathscr{C}_{\mathfrak{g}}^{0})$ are realized by the family $M(\mathfrak{C})$ for $\mathfrak{C} = (0, \mathcal{L}, \mathcal{L}, \ldots)$, and $\mathscr{C}_{\mathfrak{g}}^{0}$ provides a monoidal categorification of $K(\mathscr{C}_{\mathfrak{g}}^{0})$.
• Mutation of the seed $\mathscr{S}$ corresponds to the $T$-system, and the exact sequence (4.1) realizes the $T$-system relations via $R$-matrix homomorphisms.
• All $M(\mathfrak{C})$ are mutation equivalent, and the construction generalizes uniformly to all $[a,b] \subseteq \mathbb{Z} \sqcup \{\pm\infty\}$.