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Haiying Li

Publications and source records attributed to Haiying Li.

6 recordsLinked to original sources
arXiv2026

Symmetric Rota-Baxter systems and applications

Rota-Baxter operators and bialgebras are closely connected in several applications, such as the Connes-Kreimer renormalization framework and the operator approach to the classical Yang-Baxter equation. The concept of a Rota-Baxter system was introduced in 2016 as a generalization of a Rota-Baxter operator. In this work, we introduce a bialgebra structure compatible with a symmetric Rota-Baxter system, called a symmetric Rota-Baxter antisymmetric infinitesimal (ASI) bisystem. This bialgebra is characterized by generalizations of matched pairs of algebras and double constructions of Frobenius algebras to the setting of symmetric Rota-Baxter systems. Investigating the coboundary case leads to an enriched version of the associative Yang-Baxter equation (aYBe) adapted to symmetric Rota-Baxter systems. Antisymmetric solutions of this equation are used to construct symmetric Rota-Baxter ASI bisystems. We also introduce the notion of an $\mathcal{O}$-operator on a symmetric Rota-Baxter system, which produces solutions of the admissible aYBe in such systems and thereby gives rise to symmetric Rota-Baxter ASI bisystems. A symmetric Rota-Baxter bisystem generalizes several known structures, including Rota-Baxter Lie bisystems, Rota-Baxter ASI bialgebras, Rota-Baxter Lie bialgebras, averaging ASI bialgebras, averaging Lie bialgebras, and special apre-perm bialgebras.

math.RA↗
arXiv2025

Classical Yang-Baxter equations and Nijenhuis operators for Lie algebras

In this paper the conditions that when a Lie algebra is Nijenhuis are investigated. Furthermore all the Nijenhuis operators on $\mathfrak{sl}_2$ under the standard Cartan-Weyl basis are given. On the other hand, the relations between the classical Yang-Baxter equation and Nijenhuis operators $N$ on a Lie algebra and $P$ on a Lie coalgebra are derived by means of the bialgebraic theory for Nijenhuis Lie algebras.

math.RA↗
arXiv2024

When Leibniz algebras are Nijenhuis?

Leibniz algebras can be seen as a ``non-commutative" analogue of Lie algebras. Nijenhuis operators on Leibniz algebras introduced by Cariñena, Grabowski, and Marmo in [J. Phys. A: Math. Gen. 37(2004)] are (1, 1)-tensors with vanishing Nijenhuis torsion. Recently triangular Leibniz bialgebras were introduced by Tang and Sheng in [J. Noncommut. Geom. 16(2022)] via the twisting theory of twilled Leibniz algebras. In this paper we find that Leibniz algebras are very closely related to Nijenhuis operators, and prove that a triangular symplectic Leibniz bialgebra together with a dual triangular structure must possess Nijenhuis operators, which makes it possible to study the applications of Nijehhuis operators from the perspective of Leibniz algebras. At the same time, we regain the classical Leibniz Yang-Baxter equation by using the tensor form of classical $r$-matrics. At last we give the classification of triangular Leibniz bialgebras of low dimensions.

math.RA↗
arXiv2016

A construction of Hom-Yetter-Drinfeld category

In continuation of our recent work about smash product Hom-Hopf algebras in \cite{MLY}, we introduce Hom-Yetter-Drinfeld category $_H^H{\mathbb{YD}}$ via Radford biproduct Hom-Hopf algebra, and prove that the Hom-Yetter-Drinfeld modules can provide solutions of the Hom-Yang-Baxter equation and $_H^H{\mathbb{YD}}$ is a pre-braided tensor category, where $(H, \b, S)$ is a Hom-Hopf algebra. Furthermore, we obtain that $(A^{\natural}_{\diamond} H,\aø\b)$ is a Radford biproduct Hom-Hopf algebra if and only if $(A,\a)$ is a Hopf algebra in the category $_H^H{\mathbb{YD}}$. At last, some examples and applications are given.

math.RA↗
arXiv2016

Cobraided Smash Product Hom-Hopf Algebras

Let $(A,\a)$ and $(B,\b)$ be two Hom-Hopf algebras. In this paper, we construct a class of new Hom-Hopf algebras: $R$-smash product $(A\natural_R B,\aø\b)$. Moreover, necessary and sufficient conditions for $(A\natural_R B,\aø\b)$ to be a cobraided Hom-Hopf algebra are given.

math.RA↗