Research

2025

1. (with R. Liu, C. Zhang) Twisted combinatorial Hopf algebras and their algebraic combinatorial realizations.

   Submitted, 2025

   Abs DOI arXiv Bib

   As the BiHom-twisted deformation arises, some classical Q-deformed objects have been shown as special cases. Motivated by these, we provide a new twisted version of the combinatorial Hopf algebras, and also give their algebraic and combinatorial constructions, respectively. Specifically, we endow the BiHom-twisted structure to a graded and connected Hopf algebra, and give a proper algebraic construction of the tensor product of the twisted combinatorial Hopf algebras, which contains a proof of the statement about the tensor product of BiHom-associative algebras of G. Graziani, A. Makhlouf, C. Menini, and F. Panaite generalize the case to bialgebra setting. Applying the new twisting action, we present a combinatorial realization by reconstructing binary planar trees as an example. Finally, we conclude with some open problems for future research.

   @article{Twistcombhopf,
     title = {(with R. Liu, C. Zhang) **Twisted combinatorial Hopf algebras and their algebraic combinatorial realizations**.},
     type = {article},
     journal = {Submitted},
     doi = {},
     volume = {18},
     number = {2},
     pages = {85--92},
     year = {2025},
     publisher = {}
   }

2. (with N. Jing, Y. Liu, J. Sun, C. Zhao) On an optimal problem of bilinear forms.

   To appear in Involve, 2025

   Abs DOI arXiv Bib

   We study an optimization problem originated from the Grothendieck constant. A generalized normal equation is proposed and analyzed. We establish a correspondence between solutions of the general normal equation and its dual equation. Explicit solutions are described for the two-dimensional case.

   @article{Optimal,
     title = {(with N. Jing, Y. Liu, J. Sun, C. Zhao) **On an optimal problem of bilinear forms**.},
     type = {article},
     journal = {To appear in Involve},
     doi = {},
     volume = {18},
     number = {2},
     pages = {85--92},
     year = {2025},
     publisher = {}
   }

3. (with J. Sun, S. Wang, C. Zhang) Sweedler duality for Hom-algebras and Hom-modules.

   To appear in CAM, 2025

   Abs DOI arXiv Bib

   The construction of Sweedler duality is an important tool in the theory of Hopf algebras over a field, which is a right adjoint to the dual algebra functor. In this paper, we study the Sweedler duality of Hom-algebras and their Hom-modules. We delve into the structure of Hom-coalgebras and derive the linear morphisms associated with them. Additionally, as an application, we present the (right) Hom-(co)module morphisms under the Sweedler duality.

   @article{Sweedler,
     title = {(with J. Sun, S. Wang, C. Zhang) **Sweedler duality for Hom-algebras and Hom-modules**.},
     type = {article},
     journal = {To appear in CAM},
     doi = {},
     volume = {18},
     number = {2},
     pages = {85--92},
     year = {2025},
     publisher = {}
   }

4. (with R. Liu, C. Zhang) Deformations of Ore extensions and (p,q)-Weyl algebras.

   submitted, 2025

   Abs DOI arXiv Bib

   We obtain a BiHom deformed version of the Ore extension which takes into account the absence of a unit element and the associative properties. This gener- alises previous results of Ore and Back-Richter-Silvestrov. We also provide an exam- ple of a (p,q)-type Weyl algebra to show that the BiHom-Ore extension approach can prepare us for future realisations of new deformations of quantum algebras. As a by-product, we provide some corrections in the literature about the existence of (Bi)Hom quantum plane and (Bi)Hom universal enveloping algebras via Ore extensions.

   @article{Orepolynomial,
     title = {(with R. Liu, C. Zhang) **Deformations of Ore extensions and (p,q)-Weyl algebras**.},
     type = {article},
     journal = {submitted},
     doi = {},
     volume = {18},
     number = {2},
     pages = {85--92},
     year = {2025},
     publisher = {}
   }

2024

1. A generalization of Boole’s formula derived from a system of linear equations.

   Elemente der Mathematik, 2024

   Abs DOI arXiv Bib

   We analyze a system of linear algebraic equations whose solutions lead to a proof of a generalization of Boole’s formula. In particular, our approach provides an elementary and short alternative to Katsuura’s proof of this generalization.

   @article{Booleformula,
     title = {**A generalization of Boole's formula derived from a system of linear equations**.},
     type = {article},
     journal = {Elemente der Mathematik},
     doi = {https://doi.org/10.4171/EM/533},
     volume = {79},
     number = {4},
     pages = {173--175},
     year = {2024},
     publisher = {EMS Press}
   }

2. On an eigenvalue inequality.

   International Journal of Algebra, 2024

   Abs DOI Bib

   The eigenvalues of the sum of two matrices A and B can be bounded by Weyl’s inequality. Using the matrix decomposition technique, tighter bounds on the eigenvalues of the sum of Hermitian matrices are given with some sufficient conditions, which will be useful to compute the energy difference between quantum states or to deal with the spectrum problems.

   @article{eigenvalueinequality,
     title = {**On an eigenvalue inequality**.},
     type = {article},
     journal = {International Journal of Algebra},
     doi = {https://doi.org/10.12988/ija.2024.91859 },
     volume = {18},
     number = {2},
     pages = {85--92},
     year = {2024},
     publisher = {Hikari Press}
   }