arXiv subjects

Shiquan Ruan

Publications and source records attributed to Shiquan Ruan.

At least 37 records · Page 2Linked to original sources

arXiv2023

Geometric model for weighted projective lines of type $(p,q)$

We give a geometric model for the category of coherent sheaves over the weighted projective line of type $(p,q)$ in terms of an annulus with marked points on its boundary. We establish a bijection between indecomposable sheaves over the weighted projective line and certain homotopy classes of oriented curves in the annulus, and prove that the dimension of extension group between indecomposable sheaves equals to the positive intersection number between the corresponding curves. By using the geometric model, we provide a combinatorial description for the titling graph of tilting bundles, which is composed by quadrilaterals (or degenerated to a line). Moreover, we obtain that the automorphism group of the coherent sheaf category is isomorphic to the mapping class group of the marked annulus, and show the compatibility of their actions on the tilting graph of coherent sheaves and on the triangulation of the geometric model respectively. A geometric description of the perpendicular category with respect to an exceptional sheaf is presented at the end of the paper.

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arXiv2023

$\imath$Hall algebra of Jordan quiver and $\imath$Hall-Littlewood functions

We show that the $\imath$Hall algebra of the Jordan quiver is a polynomial ring in infinitely many generators and obtain transition relations among several generating sets. We establish a ring isomorphism from this $\imath$Hall algebra to the ring of symmetric functions in two parameters $t, θ$, which maps the $\imath$Hall basis to a class of (modified) inhomogeneous Hall-Littlewood ($\imath$HL) functions. The (modified) $\imath$HL functions admit a formulation via raising and lowering operators. We formulate and prove Pieri rules for (modified) $\imath$HL functions. The modified $\imath$HL functions specialize at $θ=0$ to the modified HL functions; they specialize at $θ=1$ to the deformed universal characters of type C, which further specialize at $(t=0, θ=1)$ to the universal characters of type C.

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arXiv2023

Quantum Borcherds-Bozec algebras via semi-derived Ringel-Hall algebras II: braid group actions

Based on the realization of quantum Borcherds-Bozec algebra $\widetilde{\mathbf{U}}$ and quantum generalized Kac-Moody algebra ${}^B\widetilde{\mathbf{U}}$ via semi-derived Ringel-Hall algebra of a quiver with loops, we deduce the braid group actions of $\widetilde{\mathbf{U}}$ introduced by Fan and Tong recently and establish braid group actions for ${}^B\widetilde{\mathbf{U}}$ by applying the BGP reflection functors to semi-derived Ringel-Hall algebras.

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arXiv2023

Stability approach to torsion pairs on abelian categories

In this paper we introduce a local-refinement procedure to investigate stability data on an abelian category, and provide a sufficient and necessary condition for a stability data to be finest. We classify all the finest stability data for the categories of coherent sheaves over certain weighted projective curves, including the classical projective line, smooth elliptic curves and certain weighted projective lines. As applications, we obtain a classification of torsion pairs for these categories via stability data approach. As a by-product, a new proof for the classification of torsion pairs in any tube category is also provided.

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arXiv2022

$\imath$quantum groups of split type via derived Hall algebras

A quantum symmetric pair consists of a quantum group $\mathbf U$ and its coideal subalgebra ${\mathbf U}^{\imath}_{\boldsymbolς}$ (called an $\imath$quantum group) with parameters ${\boldsymbolς}$. In this note, we use the derived Hall algebras of 1-periodic complexes to realize the $\imath$quantum groups ${\mathbf U}^{\imath}_{\boldsymbolς}$ of split type.

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arXiv2022

$\imath$Hall algebra of the projective line and $q$-Onsager algebra

The $\imath$Hall algebra of the projective line is by definition the twisted semi-derived Ringel-Hall algebra of the category of $1$-periodic complexes of coherent sheaves on the projective line. This $\imath$Hall algebra is shown to realize the universal $q$-Onsager algebra (i.e., $\imath$quantum group of split affine $A_1$ type) in its Drinfeld type presentation. The $\imath$Hall algebra of the Kronecker quiver was known earlier to realize the same algebra in its Serre type presentation. We then establish a derived equivalence which induces an isomorphism of these two $\imath$Hall algebras, explaining the isomorphism of the $q$-Onsager algebra under the two presentations.

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arXiv2022

New realization of $\imath$quantum groups via $Δ$-Hall algebras

For an essentially small hereditary abelian category $\mathcal{A}$, we define a new kind of algebra $\mathcal{H}_Δ(\mathcal{A})$, called the $Δ$-Hall algebra of $\mathcal{A}$. The basis of $\mathcal{H}_Δ(\mathcal{A})$ is the isomorphism classes of objects in $\mathcal{A}$, and the $Δ$-Hall numbers calculate certain three-cycles of exact sequences in $\mathcal{A}$. We show that the $Δ$-Hall algebra $\mathcal{H}_Δ(\mathcal{A})$ is isomorphic to the 1-periodic derived Hall algebra of $\mathcal{A}$. By taking suitable extension and twisting, we can obtain the $\imath$Hall algebra and the semi-derived Hall algebra associated to $\mathcal{A}$ respectively. When applied to the the nilpotent representation category $\mathcal{A}={\rm rep^{nil}}(\mathbf{k} Q)$ for an arbitrary quiver $Q$ without loops, the (\emph{resp.} extended) $Δ$-Hall algebra provides a new realization of the (\emph{resp.} universal) $\imath$quantum group associated to $Q$.

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arXiv2022

Derived equivalences between one-branch extensions of "rectangles"

In this paper we investigate the incidence algebras arising from one-branch extensions of "rectangles". There are four different ways to form such extensions, and all four kinds of incidence algebras turn out to be derived equivalent. We provide realizations for all of them by tilting complexes in a Nakayama algebra. As an application, we obtain the explicit formulas of the Coxeter polynomials for a half of Nakayama algebras (i.e., the Nakayama algebras $N(n,r)$ with $2r\geq n+2$). Meanwhile, an unexpected derived equivalence between Nakayama algebras $N(2r-1,r)$ and $N(2r-1,r+1)$ has been found.

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arXiv2022

Nakayama algebras and Fuchsian singularities

This present paper is devoted to the study of a class of Nakayama algebras $N_n(r)$ given by the path algebra of the equioriented quiver $\mathbb{A}_n$ subject to the nilpotency degree $r$ for each sequence of $r$ consecutive arrows. We show that the Nakayama algebras $N_n(r)$ for certain pairs $(n,r)$ can be realized as endomorphism algebras of tilting objects in the bounded derived category of coherent sheaves over a weighted projective line, or in its stable category of vector bundles. Moreover, we classify all the Nakayama algebras $N_n(r)$ of Fuchsian type, that is, derived equivalent to the bounded derived categories of extended canonical algebras. We also provide a new way to prove the classification result on Nakayama algebras of piecewise hereditary type, which have been done by Happel--Seidel before.

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arXiv2021

Equivariant approach to weighted projective curves

We investigate group actions on the category of coherent sheaves over weighted projective lines. We show that the equivariant category with respect to certain finite group action is equivalent to the category of coherent sheaves over a weighted projective curve, where the genus of the underlying smooth projective curve and the weight data are given by explicit formulas. Moreover, we obtain a trichotomy result for all such equivariant relations. This equivariant approach provides an effective way to investigate smooth projective curves via weighted projective lines. As an application, we give a description for the category of coherent sheaves over a hyperelliptic curve of arbitrary genus. Links to smooth projective curves of genus 2 and also to Arnold's exceptional unimodal singularities are also established.

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arXiv2020

Admissible homomorphisms and equivariant relations between weighted projective lines

The string group acts on the category of coherent sheaves over a weighted projective line by degree-shift actions. We study the equivariant equivalence relations induced by degree-shift actions between weighted projective lines. We prove that such an equivariant equivalence is characterized by an admissible homomorphism between the associated string groups. We classify all these equivariant equivalences for the weighted projective lines of domestic and tubular types.

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arXiv2020

The dual actions, equivariant autoequivalences and stable tilting objects

For a finite abelian group action on a linear category, we study the dual action given by the character group acting on the category of equivariant objects. We prove that the groups of equivariant autoequivalences on these two categories are isomorphic. In the triangulated situation, this isomorphism implies that the classifications of stable tilting objects for these two categories are in a natural bijection. We apply these results to stable tilting complexes on weighted projective lines of tubular type.

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arXiv2019

Recollements and Ladders for weighted projective lines

In this paper, we construct recollements and ladders for exceptional curves by using reduction/insertion functors due to $p$-cycle construction. As applications to weighted projective lines, we classify recollements for the category of coherent sheaves over a weighted projective line, and give an explicit description of ladders in two different levels: the bounded derived category of coherent sheaves and the stable category of vector bundles.

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arXiv2019

Tilting objects on tubular weighted projective lines: a cluster tilting approach

Using cluster tilting theory, we investigate tilting objects in the stable category of vector bundles on a weighted projective line of weight type $(2, 2, 2, 2)$. More precisely, a tilting object consisting of rank-two bundles is constructed via cluster tilting mutation. Moreover, the cluster tilting approach also provides a new method to classify the endomorphism algebras of tilting objects in the category of coherent sheaves and the associated bounded derived category.

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arXiv2019

On tubular tilting objects in the stable category of vector bundles

The present paper focuses on the study of the stable category of vector bundles for the weighted projective lines of weight triple. We find some important triangles in this category and use them to construct tilting objects with tubular endomorphism algebras for the case of genus one via cluster tilting theory.

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arXiv2018

T-stabilities for a weighted projective line

The present paper focuses on the study of t-stabilities on a triangulated category in the sense of Gorodentsev, Kuleshov and Rudakov. We give an equivalent description for the finest t-stability on a piecewise hereditary triangulated category and, describe the semistable subcategories and final HN triangles for (exceptional) coherent sheaves in $D^b(\rm{coh}\mathbb{X})$, which is the bounded derived category of coherent sheaves on the weighted projective line $\mathbb{X}$ of weight type (2). Furthermore, we show the existence of a t-exceptional triple for $D^b(\rm{coh}\mathbb{X})$. As an application, we obtain a result of Dimitrov--Katzarkov which states that each stability condition $σ$ in the sense of Bridgeland admits a $σ$-exceptional triple for the acyclic triangular quiver $Q$. Note that this implies the connectedness of the space of stability conditions associated to $Q$.

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arXiv2017

Applications of mutations in the derived categories of weighted projective lines to Lie and quantum algebras

Let $\rm{coh}\mathbb{X}$ be the category of coherent sheaves over a weighted projective line $\mathbb{X}$ and let $D^b(\rm{coh}\mathbb{X})$ be its bounded derived category. The present paper focuses on the study of the right and left mutation functors arising in $D^b(\rm{coh}\mathbb{X})$ attached to certain line bundles. As applications, we first show that these mutation functors give rise to simple reflections for the Weyl group of the star shaped quiver $Q$ associated with $\mathbb{X}$. By further dealing with the Ringel--Hall algebra of $\mathbb{X}$, we show that these functors provide a realization for Tits' automorphisms of the Kac--Moody algebra $\frak{g}_Q$ associated with $Q$, as well as for Lusztig's symmetries of the quantum enveloping algebra of ${\frak g}_Q$.

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arXiv2017

Lie algebras arising from 1-cyclic perfect complexes

Let $A$ be the path algebra of a Dynkin quiver $Q$ over a finite field, and $\mathscr{P}$ be the category of projective $A$-modules. Denote by $C^1(\mathscr{P})$ the category of 1-cyclic complexes over $\mathscr{P}$, and $\tilde{\mathfrak{n}}^+$ the vector space spanned by the isomorphism classes of indecomposable and non-acyclic objects in $C^1(\mathscr{P})$. In this paper, we prove the existence of Hall polynomials in $C^1(\mathscr{P})$, and then establish a relationship between the Hall numbers for indecomposable objects therein and those for $A$-modules. Using Hall polynomials evaluated at $1$, we define a Lie bracket in $\tilde{\mathfrak{n}}^+$ by the commutators of degenerate Hall multiplication. The resulting Hall Lie algebras provide a broad class of nilpotent Lie algebras. For example, if $Q$ is bipartite, $\tilde{\mathfrak{n}}^+$ is isomorphic to the nilpotent part of the corresponding semisimple Lie algebra; if $Q$ is the linearly oriented quiver of type $\mathbb{A}_{n}$, $\tilde{\mathfrak{n}}^+$ is isomorphic to the free 2-step nilpotent Lie algebra with $n$-generators. Furthermore, we give a description of the root systems of different $\tilde{\mathfrak{n}}^+$. We also characterize the Lie algebras $\tilde{\mathfrak{n}}^+$ by generators and relations. When $Q$ is of type $\mathbb{A}$, the relations are exactly the defining relations. As a byproduct, we construct an orthogonal exceptional pair satisfying the minimal Horseshoe lemma for each sincere non-projective indecomposable $A$-module.

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