arXiv subjects

Qinxiu Sun

Publications and source records attributed to Qinxiu Sun.

16 recordsLinked to original sources

arXiv2026

Infinite-dimensional pre-Lie bialgebras induced from Leibniz-dendriform bialgebras and Zinbiel-dendriform bialgebras

In this paper, we establish a completed pre-Lie bialgebra structure on the tensor product of a Leibniz-dendriform bialgebra and a quadratic $\mathbb{Z}$-graded Zinbiel algebra. We also obtain such a structure on the tensor product of a Zinbiel-dendriform bialgebra and a quadratic $\mathbb{Z}$-graded Leibniz algebra. Moreover, a Zinbiel-dendriform bialgebra is precisely one whose affinization by a special quadratic $\mathbb{Z}$-graded Leibniz algebra is a completed pre-Lie bialgebra. Finally, using solutions of the ZD-YBE (resp.~LD-YBE) with invariant skew-symmetric parts in a Zinbiel-dendriform (resp.~ Leibniz-dendriform) algebra, we construct completed solutions possessing invariant symmetric parts of the $S$-equation in the induced pre-Lie algebra.

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arXiv2026

Noncommutative pre-Poisson bialgebras and relative Rota-Baxter operators

In this paper, we develop the bialgebra theory for coherent noncommutative pre-Poisson algebras and establish equivalences among matched pairs, Manin triples, the phase space of noncommutative Poisson algebras and noncommutative pre-Poisson bialgebras. The investigation of coboundary noncommutative pre-Poisson bialgebras naturally leads to the noncommutative pre-Poisson Yang-Baxter equation (NPP-YBE). We prove that a symmetric solution of the NPP-YBE gives rise to a (coboundary) noncommutative pre-Poisson bialgebra. Moreover, we demonstrate how solutions without the symmetry condition can also generate such bialgebras. This motivates the introduction of quasi-triangular and factorizable noncommutative pre-Poisson bialgebras.In particular, we show that a solution of the NPP-YBE with an invariant skew-symmetric part yields a quasi-triangular noncommutative pre-Poisson bialgebra.Such solutions are further interpreted as relative Rota-Baxter operators with weights. Finally, we establish a one-to-one correspondence between quadratic Rota-Baxter noncommutative pre-Poisson algebras and factorizable noncommutative pre-Poisson bialgebras.

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arXiv2025

Leibniz-dendriform bialgebras and relative Rota-Baxter operators

In this paper, we introduce the notion of Leibniz-dendriform bialgebras and establish their equivalence with phase spaces and matched pairs of Leibniz algebras. The study of the coboundary case leads naturally to the Leibniz-dendriform Yang-Baxter equation (LD-YBE). We prove that skew-symmetric solutions of the LD-YBE give rise to coboundary Leibniz-dendriform bialgebras. Furthermore, we demonstrate that solutions not necessarily skew-symmetric can also induce such bialgebras. This observation motivates the introduction of quasi-triangular and factorizable Leibniz-dendriform bialgebras. In particular, we show that solutions of the LD-YBE with invariant symmetric parts yield quasi-triangular Leibniz-dendriform bialgebras. Such solutions are also interpreted as relative Rota-Baxter operators with weights. Finally, we establish a one-to-one correspondence between quadratic Rota-Baxter Leibniz-dendriform algebras and factorizable Leibniz-dendriform bialgebras.

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arXiv2025

FMD-TransUNet: Abdominal Multi-Organ Segmentation Based on Frequency Domain Multi-Axis Representation Learning and Dual Attention Mechanisms

Accurate abdominal multi-organ segmentation is critical for clinical applications. Although numerous deep learning-based automatic segmentation methods have been developed, they still struggle to segment small, irregular, or anatomically complex organs. Moreover, most current methods focus on spatial-domain analysis, often overlooking the synergistic potential of frequency-domain representations. To address these limitations, we propose a novel framework named FMD-TransUNet for precise abdominal multi-organ segmentation. It innovatively integrates the Multi-axis External Weight Block (MEWB) and the improved dual attention module (DA+) into the TransUNet framework. The MEWB extracts multi-axis frequency-domain features to capture both global anatomical structures and local boundary details, providing complementary information to spatial-domain representations. The DA+ block utilizes depthwise separable convolutions and incorporates spatial and channel attention mechanisms to enhance feature fusion, reduce redundant information, and narrow the semantic gap between the encoder and decoder. Experimental validation on the Synapse dataset shows that FMD-TransUNet outperforms other recent state-of-the-art methods, achieving an average DSC of 81.32\% and a HD of 16.35 mm across eight abdominal organs. Compared to the baseline model, the average DSC increased by 3.84\%, and the average HD decreased by 15.34 mm. These results demonstrate the effectiveness of FMD-TransUNet in improving the accuracy of abdominal multi-organ segmentation.

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arXiv2025

Anti-pre-Poisson bialgebras and relative Rota-Baxter operators

In this paper, we first introduce the notion of an anti-pre-Poisson bialgebra, which is shown to be equivalent to both quadratic anti-pre-Poisson algebras and matched pairs of Poisson algebras. The study of coboundary anti-pre-Poisson bialgebras leads to the anti-pre-Poisson Yang-Baxter equation (APP-YBE). Skew-symmetric solutions of this equation give rise to coboundary anti-pre-Poisson bialgebras. Furthermore, we investigate how solutions without skew-symmetry can also induce such bialgebras, prompting the introduction of quasi-triangular and factorizable anti-pre-Poisson bialgebras. In particular, solutions of the APP-YBE whose symmetric parts are invariant induce a quasi-triangular anti-pre-Poisson bialgebra. Such solutions are also interpreted as relative Rota-Baxter operators with weights. Finally, we establish a one-to-one correspondence between quadratic Rota-Baxter anti-pre-Poisson algebras and factorizable anti-pre-Poisson bialgebras.

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arXiv2025

Quasi-triangular, factorizable anti-dendriform bialgebras and relative Rota-Baxter operators

We introduce the notion of quasi-triangular anti-dendriform bialgebras, which can be induced by the solutions of the AD-YBE whose symmetric parts are invariant. A factorizable anti-dendriform bialgebra leads to a factorization of the underlying anti-dendriform algebra. Moreover, relative Rota-Baxter operators with weights are introduced to characterize the solutions of the AD-YBE whose symmetric parts are invariant. Finally, we interpret factorizable anti-dendriform bialgebras in terms of quadratic Rota-Baxter anti-dendriform algebras.

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arXiv2025

Extending structures for pre-Poisson algebras and pre-Poisson bialgebras

In this paper, we explore the extending structures problem by the unified product for pre-Poisson algebras. In particular, the crossed product and the factorization problem are investigated. Furthermore, a special case of extending structures is studied under the case of pre-Poisson algebras, which leads to the discussion of bicrossed products and matched pairs of pre-Poisson algebras. We develop a bialgebra theory for pre-Poisson algebras and establish the equivalence between matched pairs and pre-Poisson bialgebras. We study coboundary pre-Poisson bialgebras, which lead to the introduction of the pre-Poisson Yang-Baxter equation (PPYBE). A symmetric solution of the PPYBE naturally gives a coboundary pre-Poisson bialgebra.

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arXiv2025

Anti-pre-Novikov algebras, quasi-triangular and factorizable anti-pre-Novikov bialgebras

Firstly, we introduce a notion of anti-pre-Novikov algebras as a new framework for decomposing Novikov algebras. Anti-O-operators on Novikov algebras are developed to provide an algebraic framework for constructing anti-pre-Novikov algebras. Secondly, we introduce a notion of anti-pre-Novikov bialgebras as the bialgebra structures corresponding to a double construction of symmetric quasi-Frobenius Novikov algebras, which is characterized by certain matched pairs of Novikov algebras as well as the compatible anti-pre-Novikov algebras. The study of the coboundary case induces the anti-pre-Novikov Yang-Baxter equation (APN-YBE), whose skew-symmetric solutions yield coboundary anti-pre-Novikov bialgebras. The notion of O-operators on anti-pre-Novikov algebras is studied to construct skew-symmetric solutions of the APN-YBE. Thirdly, we investigate quasi-triangular and factorizable anti-pre-Novikov bialgebras as a special class of coboundary anti-pre-Novikov bialgebras. The solutions of the APN-YBE whose symmetric parts are invariant give rise to a quasi-triangular anti-pre-Novikov bialgebra. Moreover, relative Rota-Baxter operators with weights are introduced to demonstrate solutions of the APN-YBE whose symmetric parts are invariant. Finally, we introduce a notion of quadratic Rota-Baxter anti-pre-Novikov algebras, which is one to one correspondence to a factorizable anti-pre-Novikov bialgebra.

math.RA↗

arXiv2025

Extending structures for anti-dendriform algebras and anti-dendriform bialgebras

In this paper, we first explore the extending structures problem by the unified product for anti-dendriform algebras. In particular,the crossed product and non-abelian extension are studied. Furthermore, we explore the inducibility problem of pairs of automorphisms associated with a non-abelian extension of anti-dendriform algebras, and derive the fundamental sequences of Wells. Then we introduce the bicrossed products and matched pairs of anti-dendriform algebras to solve the factorization problem. Finally, we introduce the notion of anti-dendriform D-bialgebras as the bialgebra structures corresponding to double construction of associative algebras with respect to the commutative Cone cocycles. Both of them are interpreted in terms of certain matched pairs of associative algebras as well as the compatible anti-dendriform algebras. The study of coboundary cases leads to the introduction of the AD-YBE, whose skew-symmetric solutions give coboundary anti-dendriform D-bialgebras. The notion of O-operators of anti-dendriform algebras is introduced to construct skew-symmetric solutions of the AD-YBE. We also characterize the relationship between the skew-symmetric solutions of AD-YBE and O-operators.

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arXiv2024

Non-abelian extensions of Lie triple systems and Wells exact sequences

In this paper, we investigate non-abelian extensions and inducibility of pairs of automorphisms of Lie triple systems. First, we introduce non-abelian cohomology groups and classify the non-abelian extensions in terms of non-abelian cohomology groups. Next, we characterize the non-abelian extensions using Maurer-Cartan elements. Furthermore, we explore the inducibility of pairs of automorphisms and derive the analog Wells exact sequences under the circumstance of Lie triple systems. Finally, we state the previous results under the context of abelian extensions of Lie triple systems.

math.RA↗

arXiv2024

Non-abelian extensions of relative Rota-Baxter Lie algebras and Wells type exact sequences

In this paper, we explore non-abelian extensions of relative Rota-Baxter Lie algebras and classify the non-abelian extensions by introducing the non-abelian second cohomology group. We also study the inducibility of a pair of automorphisms about a non-abelian extension of relative Rota-Baxter Lie algebras and derive the Wells type exact sequences. Finally, we investigate the inducibility problem of pairs of derivations about an abelian extension of relative Rota-Baxter Lie algebras and give an exact sequence of Wells type.

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arXiv2024

Non-abelian extensions and Wells exact sequences of Lie-Yamaguti algebras

The goal of the present paper is to investigate non-abelian extensions of Lie-Yamaguti algebras and explore extensibility of a pair of automorphisms about a non-abelian extension of Lie-Yamaguti algebras. First, we study non-abelian extensions of Lie-Yamaguti algebras and classify the non-abelian extensions in terms of non-abelian cohomology groups. Next, we characterize the non-abelian extensions in terms of Maurer-Cartan elements. Moreover, we discuss the equivalent conditions of the extensibility of a pair of automorphisms about a non-abelian extension of Lie-Yamaguti algebras, and derive the fundamental sequences of Wells in the context of Lie-Yamaguti algebras. Finally, we discuss the previous results in the case of abelian extensions of Lie-Yamaguti algebras.

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arXiv2024

Cohomologies, non-abelian extensions and Wells sequences of lambda-weighted Rota-Baxter Lie coalgebras

In this paper, we investigate cohomologies and non-abelian extensions of lambda-weighted Rota-Baxter Lie coalgebras. First, we consider Lie comodules and cohomologies of lambda-weighted Rota-Baxter Lie coalgebras. Next, we study non-abelian extensions of lambda-weighted Rota-Baxter Lie coalgebras and classify the non-abelian extensions in terms of non-abelian cohomology group. Furthermore, we explore extensibility of a pair of automorphisms about a non-abelian extension of lambda-weighted Rota-Baxter Lie coalgebras, and derive the fundamental sequences of Wells in the context of lambda-weighted Rota-Baxter Lie coalgebras. Finally, we discuss the previous results in the case of abelian extensions of lambda-weighted Rota-Baxter Lie coalgebras.

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arXiv2022

Representations and cohomologies of differential 3-Lie algebras with any weight

The purpose of the present paper is to study representations and cohomologies of differential 3-Lie algebras with any weight. We introduce the representation of a differential 3-Lie algebra. Moreover,we develop cohomology theory of a differential 3-Lie algebra. We also depict the relationship between the cohomologies of a differential 3-Lie algebra and its associated differential Leibniz algebra with weight zero. The deformation and O-operator of differential 3-Lie algebras are also investigated. Finally, we consider abelian extensions of differential 3-Lie algebras.

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arXiv2022

Representations and cohomolgies of Rota-Baxter 3-Lie algebras

The goal of the present paper is to investigate representations and cohomologies of Rota-Baxter 3-Lie algebras with any weight. We introduce representations, matched pairs and Manin triples of Rota-Baxter 3-Lie algebras. Furthermore, we discuss cohomology theory of Rota-Baxter 3-Lie algebras. The deformations and central extensions of Rota-Baxter 3-Lie algebras are also studied.

math.RA↗

arXiv2020

$\mathcal{O}$-operators and related structures on Leibniz algebras

An $\mathcal{O}$-operator has been used to extend a Leibniz algebra by its representation. In this paper, we investigate several structures related to $\mathcal{O}$-operators on Leibniz algebras and introduce (dual) $\mathcal{O}$N-structures on Leibniz algebras associated to their representations. It is proved that $\mathcal{O}$-operators and dual $\mathcal{O}$N-structures generate each other under certain conditions. It is also shown that a solution of the strong Maurer-Cartan equation on the twilled Leibniz algebra gives rise to a dual $\mathcal{O}$N-structure. Finally, $r-n$ structures, RBN-structures and $\mathcal{B}N$-structures on Leibniz algebras are thoroughly studied and their interdependent relations are also studied.

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