arXiv subjects

Shiquan Ruan

Publications and source records attributed to Shiquan Ruan.

At least 19 recordsLinked to original sources

arXiv2026

Quantum supersymmetric pairs and the Serre relations via $\mathrm i$Hopf algebras

We study iHopf algebras associated with quantum supergroups of basic type. For a quantum supersymmetric pair $(\mathbf{U}, \mathbf{U}^\imath)$, we realize $\mathbf{U}^\imath$ and $\mathbf{U}$ as the iHopf algebras of the Borel quantum group ${\hat{\mathbf{U}}}_q^{\geq0}$ and the tensor product algebra ${\hat{\mathbf{U}}}_q^{\geq0}\otimes {\hat{\mathbf{U}}}_q^{\geq0}$, respectively, while the coideal subalgebra structure is encoded by an embedding between the iHopf algebras. Moreover, we derive an explicit conversion formula between the iHopf multiplication and the original multiplication, which provides a general mechanism transforming the defining relations of quantum (super)groups into relations of iquantum (super)groups. In particular, all the Serre relations that occur in quasi-split iquantum supergroups of basic type are characterized.

math.QA↗

arXiv2026

iQuantum groups and iHopf algebras I: foundation

We introduce the notion of iHopf algebra, a new associative algebra structure defined on a Hopf algebra equipped with a Hopf pairing. The iHopf algebra on a Borel quantum group endowed with a $τ$-twisted Hopf pairing is shown to be a quasi-split universal iquantum group. In particular, the Drinfeld double quantum group is realized as the iHopf algebra on the double Borel. This iHopf approach allows us to develop connections between Lusztig's braid group action and ibraid group action. It will further lead to the construction of dual canonical basis in a sequel.

math.QA↗

arXiv2026

Twisted quantum loop algebras via semi-derived Ringel-Hall algebras

Twisted quantum loop algebras are a generalization of twisted quantum affine algebras in Drinfeld new presentation. The Hall algebras of Geigle--Lenzing's weighted projective lines are used to realize (untwisted) quantum loop algebras of simply-laced type associated to star-shaped graphs by Schiffmann and Dou--Jiang--Xiao. In this paper, we use the semi-derived Ringel-Hall algebras of more general weighted projective lines to realize the twisted quantum loop algebras associated to the valued star-shaped graphs, including the twisted quantum affine algebras in Drinfeld new presentation.

math.QA↗

arXiv2026

Frobenius quotients, inflation categories and weighted projective lines

We propose the notion of Frobenius quotients between Frobenius exact categories. It turns out that any Frobenius quotient induces Frobenius quotients between the corresponding inflation categories. We obtain an explicit Frobenius quotient from the category of vector bundles on weighted projective lines with three weights to a certain category consisting of monomorphism grids.

math.CT↗

arXiv2026

Double Hall-Littlewood symmetric polynomials

We establish a ring isomorphism between the derived Hall algebra of the Jordan quiver and the ring of double symmetric functions (i.e., the ring of symmetric polynomials in two sets of countably many variables, invariant under the respective actions of their symmetric groups) with a parameter $t$. This isomorphism maps the derived Hall basis (the natural basis of the derived Hall algebra) to a class of double Hall-Littlewood (HL) symmetric functions, which are formulated via raising and lowering operators. These double HL functions are parameterized by bipartitions; they reduce to the classical HL functions when one of the partitions is empty, and specialize to Schur Laurent symmetric functions at $t = 0$. We also derive the Pieri rules for these double HL functions. Additionally, we obtain several natural generating functions for the derived Hall algebra as well as their transition relations, which can be transferred to the ring of double symmetric functions via the established ring isomorphism.

math.QA↗

arXiv2026

iQuantum groups and iHopf algebras II: dual canonical bases

Building on the iHopf algebra realization of quasi-split universal iquantum groups developed in a prequel, we construct the dual canonical basis for a universal iquantum group of arbitrary finite type, which are further shown to be preserved by the ibraid group action; this recovers the results of Lu-Pan in ADE type obtained earlier in a geometric approach. Moreover, we identify the dual canonical basis for the Drinfeld double quantum group of arbitrary finite type, which is realized via iHopf algebra on the double Borel, with Berenstein-Greenstein's double canonical basis, settling several of their conjectures.

math.QA↗

arXiv2025

ACM tilting bundles on a Geigle-Lenzing projective plane of type $(2,2,2,p)$

Let $\mathbb{X}$ be a Geigle-Lenzing projective plane of type $(2,2,2,p)$ and $\mathsf{coh} \mathbb{X}$ the category of coherent sheaves on $\mathbb{X}$. This paper is devoted to study ACM tilting bundles over $\mathbb{X}$, that is, tilting objects in the derived category $\mathsf{D}^{\rm b}(\mathsf{coh} \, \mathbb{X})$ that are also ACM bundles. We show that a tilting bundle consisting of line bundles is the $2$-canonical tilting bundle up to degree shift. We also provide a program to construct ACM tilting bundles, which give a rich source of (almost) $2$-representation infinite algebras. As an application, we give a classification result of ACM tilting bundles.

math.RT↗

arXiv2025

$\imath$Hopf algebras associated with self-dual Hopf algebras

Motivated by the construction of $\imath$Hall algebras and $Δ$-Hall algebras, we introduce $\imath$Hopf algebras associated with symmetrically self-dual Hopf algebras. We prove that the $\imath$Hopf algebra is an associative algebra with a unit, where the associativity relies on an analogue of Green's formula in the framework of Hopf algebras. As an application, we construct the $\imath$Taft algebra of dimension 4, which is proved to be isomorphic to the group algebra of $\mathbb{Z}/4\mathbb{Z}$.

math.RA↗

arXiv2025

Hall Polynomials for Weighted projective lines

This paper deals with the triangle singularity defined by the \linebreak equation $f=X_1^{p_1}+X_2^{p_2}+X_3^{p_3}$ for weight triple $(p_1,p_2,p_3)$, as well as the category of coherent sheaves over the weighted projective line $\mathbb{X}$ defined by $f$. We calculate Hall polynomials associated to extensions bundles, line bundles and torsion sheaves over $\mathbb{X}$. By using derived equivalence, this provides a unified conceptual method for calculating Hall polynomials for representations of tame quivers obtained by Szántó and Szöllősi [J. Pure Appl. Alg. {\bf 228} (2024)].

math.RT↗

arXiv2025

Analogue of Feigin's map on $\imath$quantum group of split type

The (universal) $\imath$quantum groups are as a vast generalization of (Drinfeld double) quantum groups. We establish an algebra homomorphism from universal $\imath$quantum group of split type to a certain quantum torus, which can be viewed as an $\imath$analogue of Feigin's map on the quantum group.

math.QA↗

arXiv2025

Brieskorn-Pham singularities via ACM bundles on Geigle-Lenzing projective spaces

We study the singularity category of the Brieskorn-Pham singularity $R=k[X_1, \dots, X_4]/(\sum_{i=1}^{4} X_i^{p_i})$, associated with the Geigle-Lenzing projective space $\mathbb{X}$ of weight quadruple $(p_1,\dots, p_4)$, by investigating the stable category $\underline{\mathsf{ACM}} \, \mathbb{X}$ of arithmetically Cohen-Macaulay bundles on $\mathbb{X}$. We introduce the notion of $2$-extension bundles on $\mathbb{X}$, which is a higher dimensional analog of extension bundles on a weighted projective line of Geigle-Lenzing, and then establish a correspondence between $2$-extension bundles and a certain important class of Cohen-Macaulay $R$-modules studied by Herschend-Iyama-Minamoto-Oppermann. Furthermore, we construct a tilting object in $\underline{\mathsf{ACM}} \, \mathbb{X}$ consisting of $2$-extension bundles, whose endomorphism algebra is a $4$-fold tensor product of certain Nakayama algebras. We also investigate the Picard group action on $2$-extension bundles and obtain an explicit formula for the orbit number, which gives a positive answer to a higher version of an open question raised by Kussin-Lenzing-Meltzer.

math.AG↗

arXiv2024

$\imath$Hall algebras of weighted projective lines and quantum symmetric pairs III: quasi-split type

From a category $\mathcal{A}$ with an involution $\varrho$, we introduce $\varrho$-complexes, which are a generalization of (bounded) complexes, periodic complexes and modules of $\imath$quiver algebras. The homological properties of the category $\mathcal{C}_\varrho(\mathcal{A})$ of $\varrho$-complexes are given to make the machinery of semi-derived Ringel-Hall algebras applicable. The $\imath$Hall algebra of the weighted projective line $\mathbb{X}$ is the twisted semi-derived Ringel-Hall algebra of $\mathcal{C}_\varrho({\rm coh}(\mathbb{X}))$, where $\varrho$ is an involution of ${\rm coh}(\mathbb{X})$. This $\imath$Hall algebra is used to realize the quasi-split $\imath$quantum loop algebra, which is a generalization of the $\imath$quantum group arising from the quantum symmetric pair of quasi-split affine type ADE in its Drinfeld type presentation.

math.QA↗

arXiv2024

Geometric model for vector bundles via infinite marked strips

We present a geometric model for the category of vector bundles over the weighted projective line of type (2,2,n). This model is based on the orbit space of an infinite marked strip under a specific group action. We establish a bijection between indecomposable bundles and orbits of line segments on the strip, which yields geometric interpretations for various aspects, including the Picard group action, vector bundle duality, dimension of extension group, projective cover and injective hull of extension bundle, etc.

math.RT↗

arXiv2024

$\imath$Hall algebras of weighted projective lines and quantum symmetric pairs

The $\imath$Hall algebra of a weighted projective line is defined to be the semi-derived Ringel-Hall algebra of the category of $1$-periodic complexes of coherent sheaves on the weighted projective line over a finite field. We show that this Hall algebra provides a realization of the $\imath$quantum loop algebra, which is a generalization of the $\imath$quantum group arising from the quantum symmetric pair of split affine type ADE in its Drinfeld type presentation. The $\imath$Hall algebra of the $\imath$quiver algebra of split affine type A was known earlier to realize the same algebra in its Serre presentation. We then establish a derived equivalence which induces an isomorphism of these two $\imath$Hall algebras, explaining the isomorphism of the $\imath$quantum group of split affine type A under the two presentations.

math.QA↗

arXiv2024

Derived Hall algebras of root categories

For a finitary hereditary abelian category $\mathcal{A}$, we define a derived Hall algebra of its root category by counting the triangles and using the octahedral axiom, which is proved to be isomorphic to the Drinfeld double of Hall algebra of $\mathcal{A}$. When applied to finite-dimensional nilpotent representations of the Jordan quiver or coherent sheaves over elliptic curves, these algebras provide categorical realizations of the ring of Laurent symmetric functions and also double affine Hecke algebras.

math.RT↗

arXiv2024

$\imath$Hall algebras of weighted projective lines and quantum symmetric pairs II: injectivity

We show that the morphism $Ω$ from the $\imath$quantum loop algebra $^{\texttt{Dr}}\widetilde{\mathbf{U}}(L\mathfrak{g})$ of split type to the $\imath$Hall algebra of the weighted projective line is injective if $\mathfrak{g}$ is of finite or affine type. As a byproduct, we use the whole $\imath$Hall algebra of the cyclic quiver $C_n$ to realise the $\imath$quantum loop algebra of affine $\mathfrak{gl}_n$.

math.RT↗

arXiv2024

On two open questions for extension bundles

In this paper we give positive answers for two open questions on extension bundles over weighted projective lines, raised by Kussin, Lenzing and Meltzer in the paper ``Triangle singularities, ADE-chains and weighted projective lines''.

math.AG↗

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.

math.RT↗