arXiv subjects

Masaki Kashiwara

Publications and source records attributed to Masaki Kashiwara.

At least 19 recordsLinked to original sources

arXiv2026

Monoidal seeds of the categories $\mathcal{C}_{wv}$ over quiver Hecke algebras

In this paper, when the quiver Hecke algebra R is symmetric, we present a new construction of quantum monoidal seeds $ \mathscr{S}_{w,v}$ for $\mathcal{C}_{wv}$ using the reflection functors $\mathcal{F}_i$ and the newly introduced operators $\mathcal{K}_i$. The monoidal seed $ \mathscr{S}_{w,v}$ is obtained as a subseed of the monoidal seed of $\mathcal{C}_w$ constructed by applying $\mathcal{K}_i$ and $\mathcal{F}_i$ along the special KF sequence determined by a reduced expression of $w$ and $v$. We further prove that the monoidal seed $\mathscr{S}_{w,v}$ coincides with the set of all prime factors of the determinantial modules $M(w_{\le k } Λ_{i_k}, v_{\le k} Λ_{i_k} )$. We prove that the Grothendieck ring $K(\mathcal{C}_{wv}) $ lies between the cluster algebra and the upper cluster algebra.

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arXiv2025

Faithful action of braid group on bosonic extensions

The braid group action on the bosonic extension of the quantum group has been introduced in recent works, and it can be regarded as a generalization of Lusztig's symmetries on the quantum group. In this notes, we prove the faithfulness of this braid group action.

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arXiv2025

Exchange matrices of I-boxes

Admissible chains of i-boxes are important combinatorial tools in the monoidal categorification of cluster algebras, as they provide seeds of the cluster algebra. In this paper, we explore the properties of maximal commuting families of i-boxes in a more general setting, and define a certain matrix associated with such a family, which we call the exchange matrix. It turns out that, when considering the cluster algebra structure on the Grothendieck rings, this matrix is indeed the exchange matrix of the seed associated with the family, both in certain categories of modules over quantum affine algebras and over quiver Hecke algebras. We prove this by constructing explicit short exact sequences that represent the mutation relations.

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arXiv2025

Reflection functors on quiver Hecke algebras

We construct the reflection functors for quiver Hecke algebras of an arbitrary symmetrizable Kac-Moody type. These reflection functors categorify Lusztig's braid symmetries.

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arXiv2025

Monoidal categorification and quantum affine algebras III

Let $U_q'(\mathfrak{g})$ be an arbitrary quantum affine algebra of either untwisted or twisted type, and let $\mathscr{C}_{\mathfrak{g}}^0$ be its Hernandez-Leclerc category. We denote by $\mathsf{B}$ the braid group determined by the simply-laced finite type Lie algebra $ \mathsf{g}$ associated with $U_q'(\mathfrak{g})$. For any complete duality datum $\mathbb{D}$ and any sequence of simple roots of $\mathsf{g}$, we construct the corresponding affine cuspidal modules and affine determinantial modules and study their key properties including T-systems. Then, for any element $b$ of the positive braid monoid $\mathsf{B}^+$, we introduce a distinguished subcategory $\mathscr{C}_{\mathfrak{g}}^{\mathbb{D}}(b)$ of $\mathscr{C}_{\mathfrak{g}}^0$ categorifying the specialization of the bosonic extension $\widehat{\mathcal{A}}(b)$ at $q^{1/2}=1$ and investigate its properties including the categorical PBW structure. We finally prove that the subcategory $\mathscr{C}_{\mathfrak{g}}^{\mathbb{D}}(b)$ provides a monoidal categorification of the (quantum) cluster algebra $\widehat{\mathcal{A}}(b)$, which significantly generalizes the earlier monoidal categorification developed by the authors.

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arXiv2025

Crystal Structure of Localized Quantum Unipotent Coordinate Category

A localized quantum unipotent coordinate category $\widetilde{\mathscr{C}_w}$ associated with a Weyl group element $w$ is a rigid monoidal category which is obtained by applying the localization process to a subcategory of the category of finite-dimensional graded modules of a quiver Hecke algebra. We shall show that the family of the isomorphism classes (up to grading shifts) of simple objects in $\widetilde{\mathscr{C}_w}$ possesses a crystal structure and it is isomorphic to the cellular crystal associated with $w$. As an application of this result, we shall show the connectedness of the crystal graph of an arbitrary cellular crystal.

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arXiv2024

On the irregular Riemann-Hilbert correspondence

The original Riemann-Hilbert problem asks to find a Fuchsian ordinary differential equation with prescribed singularities and monodromy in the complex line. In the early 1980's Kashiwara solved a generalized version of the problem, valid on complex manifolds of any dimension. He presented it as a correspondence between regular holonomic D-modules and perverse sheaves. The analogous problem where one drops the regularity condition remained open for about thirty years. We solved it in the paper that received a 2024 Frontiers of Science Award. Our construction requires in particular an enhancement of the category of perverse sheaves. Here, using some examples in dimension one, we wish to convey the gist of the main ingredients used in our work. This is a written account of a talk given by the first named author at the International Congress of Basic Sciences on July 2024 in Beijing.

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arXiv2024

Braid symmetries on bosonic extensions

We introduce a family of automorphisms on the bosonic extension of arbitrary type and show that they satisfy the braid relations. They preserve the global basis and the crystal basis. Using this braid group action, we define a subalgebra for each positive braid word, which possesses the PBW type basis. As an application, we show that the tensor product decomposition of the positive bosonic extionsion,

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arXiv2024

Global bases for Bosonic extensions of quantum unipotent coordinate rings

In the paper, we establish the global basis theory for the bosonic extension $\widehat{\mathcal{A}}$ associated with an arbitrary generalized Cartan matrix. When $\widehat{\mathcal{A}}$ is of simply-laced finite type, it is isomorphic to the quantum Grothendieck ring of the Hernandez-Leclerc category over a quantum affine algebra. In this case, we show that the $(t,q)$-characters of simple modules in the Hernandez-Leclerc category correspond to the normalized global basis of $\widehat{\mathcal{A}}$.

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arXiv2024

Affinizations, R-matrices and reflection functors

In this paper we establish affinizations and R-matrices in the language of pro-objects, and as an application, we construct reflection functors over the localizations of quiver Hecke algebras of arbitrary finite types. This reflection functor categorifies the braid group action on the half of a quantum group and the Saito reflection.

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arXiv2023

Localizations for quiver Hecke algebras III

Let $R$ be a quiver Hecke algebra, and let $\mathcal{C}_{w,v}$ be the category of finite-dimensional graded $R$-module categorifying a $q$-deformation of the doubly-invariant algebra $^{N'(w)} \mathbb{C}[N] ^{N(v)} $. In this paper, we prove that the localization $\tilde{\mathcal{C}}_{w,v}$ of the category $\mathcal{C}_{w,v}$ can be obtained as the localization by right braiders arising from determinantial modules. As its application, we show several interesting properties of the localized category $\tilde{\mathcal{C}}_{w,v} $ including the right rigidity.

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arXiv2023

Laurent family of simple modules over quiver Hecke algebra

We introduce the notions of quasi-Laurent and Laurent families of simple modules over quiver Hecke algebras of arbitrary symmetrizable types. We prove that such a family plays a similar role of a cluster in the quantum cluster algebra theory and exhibits a quantum Laurent positivity phenomenon for the basis of the quantum unipotent coordinate ring $\mathcal{A}_q(\mathfrak{n}(w))$, coming from the categorification. Then we show that the families of simple modules categorifying GLS-clusters are Laurent families by using the PBW-decomposition vector of a simple module $X$ and categorical interpretation of (co-)degree of $[X]$. As applications of such $\mathbb{Z}$-vectors, we define several skew symmetric pairings on arbitrary pairs of simple modules, and investigate the relationships among the pairings and $Λ$-invariants of R-matrices in the quiver Hecke algebra theory.

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arXiv2023

The $(q,t)$-Cartan matrix specialized at $q=1$

The $(q,t)$-Cartan matrix specialized at $t=1$, usually called the quantum Cartan matrix, has deep connections with (i) the representation theory of its untwisted quantum affine algebra, and (ii) quantum unipotent coordinate algebra, root system and quantum cluster algebra of kew-symmetric type. In this paper, we study the $(q,t)$-Cartan matrix specialized at $q=1$, called the $t$-quantized Cartan matrix, and investigate the relations with (ii') its corresponding quantum unipotent coordinate algebra, root system and quantum cluster algebra of skew-symmetrizable type.

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arXiv2023

$t$-quantized Cartan matrix and R-matrices for cuspidal modules over quiver Hecke algebras

As every simple module of a quiver Hecke algebra appears as the image of the R-matrix defined on the convolution product of certain cuspidal modules, knowing the $\mathbb{Z}$-invariants of the R-matrices between cuspidal modules is quite significant. In this paper, we prove that the $(q,t)$-Cartan matrix specialized at $q=1$ of an arbitrary finite type, called the $t$-quantized Cartan matrix, informs us of the invariants of R-matrices. To prove this, we use combinatorial AR-quivers associated with Dynkin quivers and their properties as crucial ingredients.

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arXiv2022

Localizations for quiver Hecke algebras II

We prove that the localization of the monoidal category $\mathcal{C}_w$ is rigid, and the category $\mathcal{C}_{w,v}$ admits a localization via a real commuting family of central objects. Note that the localization of $\mathcal{C}_{w,v}$ categorifies the open Richardson variety.

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arXiv2022

Monoidal categorification and quantum affine algebras II

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.

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arXiv2021

Categorical crystals for quantum affine algebras

A new categorical crystal structure for the quantum affine algebras is presented. We introduce the extended crystal $\widehat{B}_{\mathfrak{g}}(\infty)$ for an arbitrary quantum group, which is the product of infinite copies of the crystal $B(\infty)$. For a complete duality datum in the Hernandez-Leclerc category $\mathcal{C}^0_{\mathfrak{g}}$ of a quantum affine algebra $U_q'(\mathfrak{g})$, we prove that the set of the isomorphism classes of simple modules in $\mathcal{C}^0_{\mathfrak{g}}$ has an extended crystal structure isomorphic to the extended crystal $\widehat{B}_{\mathfrak{g}}(\infty)$. An explicit combinatorial description of the extended crystal $\widehat{B}_{\mathfrak{g}}(\infty)$ for affine type $A_n^{(1)}$ is given in terms of affine highest weights.

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arXiv2021

Simply-laced root systems arising from quantum affine algebras

Let $U_q'(\mathfrak{g})$ be a quantum affine algebra with an indeterminate $q$ and let $\mathscr{C}_{\mathfrak{g}}$ be the category of finite-dimensional integrable $U_q'(\mathfrak{g})$-modules. We write $\mathscr{C}_{\mathfrak{g}}^0$ for the monoidal subcategory of $\mathscr{C}_{\mathfrak{g}}$ introduced by Hernandez-Leclerc. In this paper, we associate a simply-laced finite type root system to each quantum affine algebra $U_q'(\mathfrak{g})$ in a natural way, and show that the block decompositions of $\mathscr{C}_{\mathfrak{g}}$ and $\mathscr{C}_{\mathfrak{g}}^0$ are parameterized by the lattices associated with the root system. We first define a certain abelian group $\mathcal{W}$ (resp. $\mathcal{W}_0$) arising from simple modules of $ \mathscr{C}_{\mathfrak{g}}$ (resp. $\mathscr{C}_{\mathfrak{g}}^0$) by using the invariant $Λ^\infty$ introduced in the previous work by the authors. The groups $\mathcal{W}$ and $\mathcal{W}_0$ have the subsets $Δ$ and $Δ_0$ determined by the fundamental representations in $ \mathscr{C}_{\mathfrak{g}}$ and $\mathscr{C}_{\mathfrak{g}}^0$ respectively. We prove that the pair $( \mathbb{R} \otimes_\mathbb{Z} \mathcal{W}_0, Δ_0)$ is an irreducible simply-laced root system of finite type and the pair $( \mathbb{R} \otimes_\mathbb{Z} \mathcal{W}, Δ) $ is isomorphic to the direct sum of infinite copies of $( \mathbb{R} \otimes_\mathbb{Z} \mathcal{W}_0, Δ_0)$ as a root system.

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