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Liangang Peng

Publications and source records attributed to Liangang Peng.

12 recordsLinked to original sources
arXiv2026

On endomorphism algebras of silting complexes over hereditary abelian categories

Let $\mathcal{E}$ be the class of finite-dimensional algebras isomorphic to endomorphism algebras of silting complexes over hereditary abelian categories. It is proved that the class $\mathcal{E}$ is closed under taking idempotent quotients, idempotent subalgebras and $\tau$-reduction. We also show that the proper class consisting of shod algebras is also closed under these operations. In addition, several classic classes of algebras -- including laura, glued, weakly shod algebras -- are proved to be closed under idempotent quotients, thereby generalizing a known result originally established for specific idempotents.

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arXiv2024

Semi-derived Ringel-Hall bialgebras

Let $\mathcal{A}$ be an arbitrary hereditary abelian category. Lu and Peng defined the semi-derived Ringel-Hall algebra $SH(\mathcal{A})$ of $\mathcal{A}$ and proved that $SH(\mathcal{A})$ has a natural basis and is isomorphic to the Drinfeld double Ringel-Hall algebra of $\mathcal{A}$. In this paper, we introduce a coproduct formula on $SH(\mathcal{A})$ with respect to the basis of $SH(\mathcal{A})$ and prove that this coproduct is compatible with the product of $SH(\mathcal{A})$, thereby the semi-derived Ringel-Hall algebra of $\mathcal{A}$ is endowed with a bialgebra structure which is identified with the bialgebra structure of the Drinfeld double Ringel-Hall algebra of $\mathcal{A}$.

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arXiv2024

Finite dimensional 2-cyclic Jacobian algebras

In this paper, we start with a class of quivers that containing only 2-cycles and loops, referred to as 2-cyclic quivers. We prove that there exists a potential on these quivers that ensures the resulting quiver with potential is Jacobian-finite. As an application, we first demonstrate, using covering theory, that a Jacobian-finite potential exists on a class of 2-acyclic quivers. Secondly, by using the 2-cyclic Caldero-Chapoton formula, the $\tau$-rigid modules over the Jacobian algebras of our proven Jacobian-finite 2-cyclic quiver with potential can categorify Paquette-Schiffler's generalized cluster algebras in three specific cases: one for a disk with two marked points and one 3-puncture, one for a sphere with one puncture, one 3-puncture and one orbifold point, and another for a sphere with one puncture and two 3-punctures.

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arXiv2024

On $F$-Polynomials for Generalized Quantum Cluster Algebras and Gupta's Formula

We show the polynomial property of $F$-polynomials for generalized quantum cluster algebras and obtain the associated separation formulas under a mild condition. Along the way, we obtain Gupta's formulas of $F$-polynomials for generalized quantum cluster algebras. These formulas specialize to Gupta's formulas for quantum cluster algebras and cluster algebras respectively. Finally, a generalization of Gupta's formula has also been discussed in the setting of generalized cluster algebras.

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arXiv2023

Semi-derived Ringel-Hall algebras and Hall algebras of odd-periodic relative derived categories

Let $t$ be a positive integer and $\mathcal{A}$ a hereditary abelian category satisfying some finiteness conditions. We define the semi-derived Ringel-Hall algebra of $\mathcal{A}$ from the category $\mathcal{C}_{\mathbb{Z}/t}(\mathcal{A})$ of $\mathbb{Z}/t$-graded complexes and obtain a natural basis of the semi-derived Ringel-Hall algebra. Moreover, we describe the semi-derived Ringel-Hall algebra by the generators and defining relations. In particular, if $t$ is an odd integer, we show that there is an embedding of derived Hall algebra of the odd-periodic relative derived category in the extended semi-derived Ringel-Hall algebra.

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arXiv2022

Quantum cluster characters of Hall algebras revisited

Let $Q$ be a finite acyclic valued quiver. We define a bialgebra structure and an integration map on the Hall algebra associated to the morphism category of projective representations of $Q$. As an application, we recover the surjective homomorphism defined in \cite{DXZ}, which realizes the principal coefficient quantum cluster algebra $\A_q(Q)$ as a sub-quotient of the Hall algebra of morphisms. Moreover, we also recover the quantum Caldero--Chapoton formula, as well as some multiplication formulas between quantum Caldero--Chapoton characters.

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arXiv2021

Quantized cohomological Hall algebra of the $d$-loop quiver revisited

Let $Λ$ be the set of partitions of length $\geq 0$. We introduce an $\mathbb{N}$-graded algebra $\mathbb{A}_q^d(Λ)$ associated to $Λ$, which can be viewed as a quantization of the algebra of partitions defined by Reineke. The multiplication of $\mathbb{A}^d_q(Λ)$ has some kind of quasi-commutativity, and the associativity comes from combinatorial properties of certain polynomials appeared in the quantized cohomological Hall algebra $\mathcal{H}^d_q$ of the $d$-loop quiver. It turns out that $\mathbb{A}^d_q(Λ)$ is isomorphic to $\mathcal{H}^d_q$, thus can be viewed as a combinatorial realization for $\mathcal{H}^d_q$.

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arXiv2021

Semi-derived Ringel-Hall algebras and Drinfeld double

Let $\mathcal{A}$ be an arbitrary hereditary abelian category that may not have enough projective objects. For example, $\mathcal{A}$ can be the category of finite-dimensional representations of a quiver or the category of coherent sheaves on a smooth projective curve or on a weighted projective line. Inspired by the works of Bridgeland and Gorsky, we define the semi-derived Ringel-Hall algebra of $\mathcal{A}$, denoted by $\mathcal{S}\mathcal{D}\mathcal{H}_{\mathbb{Z}/2}(\mathcal{A})$, to be the localization of a quotient algebra of the Ringel-Hall algebra of the category of $\mathbb{Z}/2$-graded complexes over $\mathcal{A}$. We obtain the following three main results. The semi-derived Ringel-Hall algebra has a natural basis. A twisted version of the semi-derived Ringel-Hall algebra of $\mathcal{A}$ is isomorphic to the Drinfeld double of the twisted extended Ringel-Hall algebra $\mathcal{H}_{tw}^e(\mathcal{A})$ of $\mathcal{A}$. If $\mathcal{A}$ has a tilting object $T$, then its semi-derived Ringel-Hall algebra is isomorphic to the $\mathbb{Z}/2$-graded semi-derived Hall algebra $\mathcal{S}\mathcal{D}\mathcal{H}_{\mathbb{Z}/2}(\mathrm{add} T)$ of the exact category $\mathrm{add} T$ defined by Gorsky, and so is isomorphic to Bridgeland's Hall algebra of $\mod (\mathrm{End}(T)^{op})$.

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arXiv2018

Modified Ringel-Hall Algebras, Green's formula and Derived Hall Algebras

In this paper we define the modified Ringel-Hall algebra $\cm\ch(\ca)$ of a hereditary abelian category $\ca$ from the category $C^b(\mathcal{A})$ of bounded $\mathbb{Z}$-graded complexes. Two main results have been obtained. One is to give a new proof of Green's formula on Ringel-Hall numbers by using the associative multiplication of the modified Ringel-Hall algebra. The other is to show that in certain twisted cases the derived Hall algebra can be embedded in the modified Ringel-Hall algebra. As a consequence of the second result, we get that in certain twisted cases the modified Ringel-Hall algebra is isomorphic to the tensor algebra of the derived Hall algebra and the torus of acyclic complexes and so the modified Ringel-Hall algebra is invariant under derived equivalences.

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arXiv2013

Quantum dilogarithm identities and cyclic quivers

We study quantum dilogarithm identities for cyclic quivers following Reineke's idea via Ringel-Hall algebra approach. For any given discrete stability function for the cyclic quiver $Δ_n$ with $n$ vertices, we obtain certain cyclic quantum dilogarithm identities of order $n$ in the sense of Bytsko and Volkov.

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arXiv2009

Symmetrizable intersection matrices and their root systems

In this paper we study symmetrizable intersection matrices, namely generalized intersection matrices introduced by P. Slodowy such that they are symmetrizable. Every such matrix can be naturally associated with a root basis and a Weyl root system. Using $d$-fold affinization matrices we give a classification, up to braid-equivalence, for all positive semi-definite symmetrizable intersection matrices. We also give an explicit structure of the Weyl root system for each $d$-fold affinization matrix in terms of the root system of the corresponding Cartan matrix and some special null roots.

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