arXiv subjects

Jiepeng Fang

Publications and source records attributed to Jiepeng Fang.

12 recordsLinked to original sources

arXiv2026

Thickening realization and positivity properties of canonical bases

Let $\mathbf{U}$ be a quantum group associated with a symmetric Cartan datum, let $\dot{\mathbf{U}}$ be its modified form, and let $\dot{\mathbf{B}}$ be the canonical basis of $\dot{\mathbf{U}}$. Lusztig conjectured that the structure constants of the multiplication, comultiplication, and bilinear form in $\dot{\mathbf{U}}$ with respect to $\dot{\mathbf{B}}$ belong to $\mathbb{N}[v,v^{-1}]$. We introduce the \emph{thickening realization}, which relates $\dot{\mathbf{B}}$ to the canonical basis of the negative part of a larger quantum group $\tilde{\mathbf{U}}^-$. More precisely, it identifies the relevant structure constants in $\dot{\mathbf{U}}$ with the structure constants in $\tilde{\mathbf{U}}^-$. As consequences, we prove, for arbitrary symmetric Cartan datum, $\dot{\mathbf{B}}$ has the positivity properties for the comultiplication and bilinear form, as well as the positivity for the multiplication whenever one factor is spherical parabolic. In particular, Lusztig's conjecture holds for simply-laced finite type. We also prove the canonical bases of a broad class of tensor products of integrable modules have the positivity properties for the transition matrices and actions by $\dot{\mathbf{B}}$.

math.QA↗

arXiv2026

Canonical bases of tensor products and positivity properties

Let $\mathbf{U}$ be a quantum group of symmetric type. We introduce the {\it thickening realization} to realize (a suitable approximation of) the tensor product ${^ωΛ_{λ_1}}\otimes Λ_{λ_2}$ of a simple integrable lowest weight module and a highest weight module as a subquotient of the Verma module of a larger quantum group $\tilde{\mathbf{U}}$. For the canonical basis of the tensor product, we show that the entries of the transition matrix from the pure tensor basis to it, and the structure constants of the action by spherical parabolic subalgebras of the modified quantum group $\dot{\mathbf{U}}$ are given by the structure constants of the comultiplication and multiplication in the negative part $\tilde{\mathbf{U}}^-$ of $\tilde{\mathbf{U}}$ with respect to its canonical basis respectively. Thus, we deduce the positivity property of the canonical basis of the tensor product. In particular, we obtain the positivity property of the canonical bases for the action of $\dot{\mathbf{U}}$ on simple integrable highest weight modules, generalizing Lusztig's theorem from Chevalley generators to any canonical basis elements of $\dot{\mathbf{U}}$; for the action of Chevalley generators on ${^ωΛ_{λ_1}}\otimes Λ_{λ_2}$; and for multiplication in $\dot{\mathbf{U}}$, as well as the actions on arbitrary tensor products. At $v=1$, these results connect to geometric total positivity on double flag varieties, explored in the joint work of He and Xie.

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arXiv2025

Canonical bases of tensor products of integrable highest weight modules arising from framed constructions

Given a quantum group, we prove that the canonical bases of the tensor products of its integrable highest weight modules can be obtained from the canonical bases of the integrable highest weight modules of a bigger quantum group. As a result, based on the positivity of the canonical bases of the integrable highest weight modules due to Lusztig, we prove that the canonical bases of the tensor products have the positivity.

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arXiv2025

Sheaf realization of Bridgeland's Hall algebra of Dynkin type

As one of results in [6], Bridgeland realized the quantum group $\mathbf{U}_v$ via the localization of Ringel-Hall algebra for the two-periodic projective complexes of quiver representations over a finite field. In the present paper, we generalize Lusztig's categorical construction for the nilpotent part $\mathbf{U}_v^+$ to Bridgeland's Hall algebra of Dynkin type. In particular, we obtain a basis of the Ringel-Hall algebra for the two-periodic projective complexes which has the positivity, and we categorify an integral form of the generic Bridgeland's Hall algebra which is isomorphic to the Poisson integral form of $\mathbf{U}_v$, and obtain a $\mathbb{Z}[v,v^{-1}]$-basis of this integral form.

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arXiv2025

Lusztig sheaves and tensor products of integrable highest weight modules

By introducing $N$-framed quivers, we define the localization of Lusztig's sheaves for $N$-framed quivers and functors $E^{(n)}_{i}, F^{(n)}_{i}, K^{\pm}_i$ for localizations. This gives a categorical realization of tensor products of integrable highest weight modules of the quantized enveloping algebra. The simple perverse sheaves in the localization provide a basis of the tensor product. We prove that this basis coincides with the canonical basis of tensor product in the sense of Lusztig and Bao-Wang. Moreover, we give a categorical interpretation of the Yang-Baxter equation.

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arXiv2025

Lusztig sheaves, characteristic cycles and the Borel-Moore homology of Nakajima's quiver varieties

By using characteristic cycles, we build a morphism from the canonical bases of integrable highest weight modules of quantum groups to the top Borel-Moore homology groups of Nakajima's quiver and tensor product varieties, and compare the canonical bases and the fundamental classes. As an application, we show that Nakajima's realization of irreducible highest weight modules and their tensor products can be defined over integers. We also give a new proof of Nakajima's conjecture on the canonical isomorphism of tensor product varieties.

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arXiv2025

Lusztig sheaves and integrable highest weight modules

We consider the localization $\mathcal{Q}_{\mathbf{V},\mathbf{W}}/\mathcal{N}_{\mathbf{V}}$ of Lusztig's sheaves for framed quivers, and define functors $E^{(n)}_{i},F^{(n)}_{i},K^{\pm}_{i},n\in \mathbb{N},i \in I$ between the localizations. With these functors, the Grothendieck group of localizations realizes the irreducible integrable highest weight modules $L(Λ)$ of quantum groups. Moreover, the nonzero simple perverse sheaves in localizations form the canonical bases of $L(Λ)$. We also compare our realization (at $v \rightarrow 1$) with Nakajima's realization via quiver varieties and prove that the transition matrix between canonical bases and fundamental classes is upper triangular with diagonal entries all equal to $\pm 1$.

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arXiv2024

Lie algebras arising from two-periodic projective complex and derived categories

Let $A$ be a finite-dimensional $\mathbb{C}$-algebra of finite global dimension and $\mathcal{A}$ be the category of finitely generated right $A$-modules. By using of the category of two-periodic projective complexes $\mathcal{C}_2(\mathcal{P})$, we construct the motivic Bridgeland's Hall algebra for $\mathcal{A}$, where structure constants are given by Poincaré polynomials in $t$, then construct a $\mathbb{C}$-Lie subalgebra $\mathfrak{g}=\mathfrak{n}\oplus \mathfrak{h}$ at $t=-1$, where $\mathfrak{n}$ is constructed by stack functions about indecomposable radical complexes, and $\mathfrak{h}$ is by contractible complexes. For the stable category $\mathcal{K}_2(\mathcal{P})$ of $\mathcal{C}_2(\mathcal{P})$, we construct its moduli spaces and a $\mathbb{C}$-Lie algebra $\tilde{\mathfrak{g}}=\tilde{\mathfrak{n}}\oplus \tilde{\mathfrak{h}}$, where $\tilde{\mathfrak{n}}$ is constructed by support-indecomposable constructible functions, and $\tilde{\mathfrak{h}}$ is by the Grothendieck group of $\mathcal{K}_2(\mathcal{P})$. We prove that the natural functor $\mathcal{C}_2(\mathcal{P})\rightarrow \mathcal{K}_2(\mathcal{P})$ together with the natural isomorphism between Grothendieck groups of $\mathcal{A}$ and $\mathcal{K}_2(\mathcal{P})$ induces a Lie algebra isomorphism $\mathfrak{g}\cong\tilde{\mathfrak{g}}$. This makes clear that the structure constants at $t=-1$ provided by Bridgeland in [5] in terms of exact structure of $\mathcal{C}_2(\mathcal{P})$ precisely equal to that given in [30] in terms of triangulated category structure of $\mathcal{K}_2(\mathcal{P})$.

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arXiv2024

The parity of Lusztig's restriction functor and Green's formula for a quiver with automorphism

In [8], Fang-Lan-Xiao proved a formula about Lusztig's induction and restriction functors which can induce Green's formula for the path algebra of a quiver over a finite field via the trace map. In this paper, we generalize their formula to that for the mixed semisimple perverse sheaves for a quiver with an automorphism. By applying the trace map, we obtain Green's formula for any finite-dimensional hereditary algebra over a finite field.

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arXiv2024

On singular supports of Lusztig's perverse sheaves

We prove a conjecture of Lusztig on a microlocal characterization of his perverse sheaves. For any finite quiver without loops, an equivariant simple perverse sheaf on the variety of quiver representations is a Lusztig's perverse sheaf if and only if its singular support is contained in Lusztig's Lagrangian variety, that is, the variety of nilpotent representations of the preprojective algebra of the quiver.

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arXiv2023

The correspondence between the canonical and semicanonical bases

Given any symmetric Cartan datum, Lusztig has provided a pair of key lemmas to construct the perverse sheaves over the corresponding quiver and the functions of irreducible components over the corresponding preprojective algebra respectively. In the present article, we prove that these two inductive algorithms of Lusztig coincide. Consequently we can define two colored graphs and prove that they are isomorhic. This result finishes the statement that Lusztig's functions of irreducible components are basis of the enveloping algebra and deduces the crystal structure (in the sense of Kashiwara-Saito) from the semicanonical basis directly inside Lusztig's convolution algebra of the preprojective algebra. As an application, we prove that the transition matrix between the canonical basis and the semicanonical basis is upper triangular with all diagonal entries equal to 1.

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arXiv2022

The Parity of Lusztig's Restriction Functor and Green's Formula

Our investigation in the present paper is based on three important results. (1) In [12], Ringel introduced Hall algebra for representations of a quiver over finite fields and proved the elements corresponding to simple representations satisfy the quantum Serre relation. This gives a realization of the nilpotent part of quantum group if the quiver is of finite type. (2) In [4], Green found a homological formula for the representation category of the quiver and equipped Ringel's Hall algebra with a comultiplication. The generic form of the composition subalgebra of Hall algebra generated by simple representations realizes the nilpotent part of quantum group of any type. (3) In [9], Lusztig defined induction and restriction functors for the perverse sheaves on the variety of representations of the quiver which occur in the direct images of constant sheaves on flag varieties, and he found a formula between his induction and restriction functors which gives the comultiplication as algebra homomorphism for quantum group. In the present paper, we prove the formula holds for all semisimple complexes with Weil structure. This establishes the categorification of Green's formula.

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