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Junyang Liu

Publications and source records attributed to Junyang Liu.

11 recordsLinked to original sources
arXiv2026

Triangle equivalences between Gorenstein tiled orders and incidence algebras of posets

We prove that for any $\mathbb{N}$-graded Gorenstein tiled order $A$, the stable category $\underline{\mathrm{CM}}^{\mathbb{Z}}A$ is triangle equivalent to the perfect derived category of the incidence algebra of a finite poset $\mathbb{V}_A^{op}$. Moreover, for a finite poset $P$, we prove that the incidence algebra of $P$ can be realized as the endomorphism algebra of a standard tilting object if and only if $P$ is either empty or has the maximum. We also study the behaviors of the corresponding poset under graded Morita equivalences and coverings of a Gorenstein tiled order. Finally, we classify Gorenstein tiled orders $A$ satisfying $|\mathbb{V}_A^{op}|\leq 3$.

math.RT↗
arXiv2025

Calabi-Yau structures on derived and singularity categories of symmetric orders

We construct left and right Calabi-Yau structures on derived respectively singularity categories of symmetric orders $\Lambda$ over commutative Gorenstein rings $R$. For this, we first construct Calabi-Yau structures over $R$ by lifting Amiot's construction of Calabi-Yau structures on Verdier quotients to the dg level. Then we prove base change properties relating Calabi-Yau structures over $R$ to those over the base field $k$. As a result, we prove the existence of a right Calabi-Yau structure on the dg singularity category associated with $\Lambda$ which is a cyclic lift of the weak Calabi-Yau structure constructed by the first-named author and Iyama. We also show the existence of a left Calabi-Yau structure on the dg bounded derived category of $\Lambda$. This is a non-commutative generalization of a result by Brav and Dyckerhoff. By combining the existence of the right Calabi-Yau structure on the dg singularity category with a structure theorem by Keller and the second-named author, we deduce that under suitable hypotheses, the singularity category associated with $\Lambda$ is triangle equivalent to a generalized cluster category in the sense of Amiot.

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arXiv2025

Non-Asymptotic Performance Analysis of DOA Estimation Based on Real-Valued Root-MUSIC

This paper presents a systematic theoretical performance analysis of the Real-Valued root-MUSIC (RV-root-MUSIC) algorithm under non-asymptotic conditions. A well-known limitation of RV-root-MUSIC is the estimation ambiguity caused by mirror roots, which are typically suppressed using conventional beamforming (CBF). By leveraging the equivalent subspace constructed through the conjugate extension method and exploiting the equivalence of perturbations for true and mirror roots, this work provides a comprehensive study of three key aspects: noise subspace perturbation, true-root perturbation, and mirror-root perturbation. A statistical model is established, and generalized perturbation expressions are derived. Monte Carlo simulations confirm the correctness and effectiveness of the theoretical results. The analysis provides a rigorous foundation for parameter optimization in Direction-of-Arrival (DOA) estimation, with applications in radar, wireless communications, and intelligent sensing.

cs.PF↗
arXiv2025

Tilting theory for hypersurface singularities of dimension one

Any $\mathbb{N}$-graded commutative Gorenstein ring $R$ of Krull dimension one with $R_0$ a field admits a standard silting object $V$ in the stable category $\underline{\mathrm{CM}}_0^{\mathbb{Z}}R$, and the object $V$ is tilting if and only if the $a$-invariant $a$ is non-negative, as shown by Buchweitz, the first author, and Yamaura. In this article, under the additional assumption that $R$ is a hypersurface singularity, we prove that endomorphism algebra of $V$ is Iwanaga-Gorenstein of self-injective dimension at most $2$, and we give its explicit presentation in terms of a quiver with relation. In the case of where $a$ is negative, we prove that the dg endomorphism algebra of $V$ is Gorenstein, and we give its explicit presentation in terms of a dg path algebra. We explain our results by several examples including numerical semigroup algebras generated by two elements. Moreover, for each finite and countable Cohen-Macaulay representation type, we include the Auslander-Reiten quiver of the category $\mathrm{CM}_0^{\mathbb{Z}}R$ with the position of the standard silting object. As a step of the proof of our results, we give a characterization of Gorensteinness of homologically finite dg algebras in terms of Serre functors.

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arXiv2025

Bright squeezed light in the kilohertz frequency band

The largely technical noise of a free running laser is the fundamental limit for preparation of a bright squeezed light, especially within MHz band. We construct a universal and complete theoretical model for nonclassical stabilization, and propose a novel bright squeezed light generation scheme with assistance of a hybrid noise stabilization technology. The scheme assimilates a broadband passive noise stabilization to successfully extend the feedback frequency bandwidth of a nonclassical active stabilization to MHz band. Meanwhile, the technical noise suppression magnitude is further improved by 9 dB, which creates the prerequisite conditions for preparing a broadband bright squeezed light. Finally, a -5.5 dB bright squeezed light with a power of 1 mW and a squeezing bandwidth among kHz to MHz band was generated. The experimental results agree well with the theoretical one. To the best of our knowledge, it is the first demonstration of a milliwatt-order bright squeezed light across the kHz frequency band. The demonstration sheds light on potential applications of bright squeezed light in quantum metrology and might enable new concepts in the future.

quant-ph↗
arXiv2024

Singularity categories via higher McKay quivers with potential

In 2018, Kalck and Yang showed that the singularity categories associated with $3$-dimensional Gorenstein quotient singularities are triangle equivalent (up to direct summands) to small cluster categories associated with McKay quivers with potential. We introduce higher McKay quivers with potential and generalize Kalck and Yang's theorem to arbitrary dimensions. The singularity categories we consider occur as the stable categories of categories of Cohen-Macaulay modules. We refine our description of the singularity categories by showing that these categories of Cohen-Macaulay modules are equivalent to Higgs categories in the sense of Wu. Moreover, we describe the singularity categories in the non-Gorenstein case.

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arXiv2023

On Amiot's conjecture

In a survey paper in 2011, Amiot proposed a conjectural characterisation of the cluster categories which were conceived in the mid 2000s to lift the combinatorics of Fomin-Zelevinsky's cluster algebras to the categorical level. This paper is devoted to a proof of (a variant of) her conjecture. More generally, cluster categories admit higher-dimensional and relative variants, the so-called Higgs categories recently introduced by Wu. We also prove higher-dimensional and relative variants of the conjecture.

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arXiv2023

Relative Calabi-Yau structures and ice quivers with potential

In 2015, Van den Bergh showed that complete $3$-Calabi-Yau algebras over an algebraically closed field of characteristic $0$ are equivalent to Ginzburg dg algebras associated with quivers with potential. He also proved the natural generalisation to higher dimensions and non-algebraically closed ground fields. The relative version of the notion of Ginzburg dg algebra is that of Ginzburg morphism. For example, every ice quiver with potential gives rise to a Ginzburg morphism. We generalise Van den Bergh's theorem by showing that, under suitable assumptions, any morphism with a relative Calabi-Yau structure is equivalent to a Ginzburg(-Lazaroiu) morphism. In particular, in dimension $3$ and over an algebraically closed ground field of characteristic $0$, it is given by an ice quiver with potential. Thanks to the work of Bozec-Calaque-Scherotzke, this result can also be viewed as a noncommutative analogue of Joyce-Safronov's Lagrangian neighbourhood theorem in derived symplectic geometry.

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arXiv2023

Calabi-Yau structures on Drinfeld quotients and Amiot's conjecture

In 2009, Claire Amiot gave a construction of Calabi-Yau structures on Verdier quotients. We sketch how to lift it to the dg setting. We use this construction as an important step in an outline of the proof of her conjecture on the structure of 2-Calabi-Yau triangulated categories with a cluster-tilting object.

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arXiv2023

$A_\infty$-deformations of zigzag algebras via Ginzburg dg algebras

This note aims to give a short proof of the recent result due to Etg\"u-Lekili (2017) and Lekili-Ueda (2021): the zigzag algebra of any finite tree over a field of characteristic 0 is intrinsically formal if and only if the tree is of type ADE. We also complete the proof of this result by considering a field of arbitrary characteristic for type E, which was still open.

math.RT↗