12 citations · 119 across the 39 of their papers we have counts for
41 papers
Free weighted differential (-tri)dendriform algebras
Yuanyuan Zhang, Huhu Zhang, Tingzeng Wu +1
In the present paper, we propose the concepts of weighted differential (-tri)dendriform algebras and give some basic properties of them. The corresponding free objects are const…
A Poincaré-Birkhoff-Witt theorem for the universal enveloping algebra of a Rota-Baxter Lie algebra
Zhi-Cheng Zhu, Xing Gao, Li Guo +1
Rota-Baxter associative algebras and Rota-Baxter Lie algebras are both important in mathematics and mathematical physics, with the former a basic structure in quantum field renorma…
Free -Rota-Baxter systems and Gröbner-Shirshov bases
Yuanyuan Zhang, Huhu Zhang, Xing Gao
In this paper, we propose the concept of an -Rota-Baxter system, which is a generalization of a Rota-Baxter system and an -Rota-Baxter algebra of weight zero. In the framewor…
Operated groups, differential groups and Rota-Baxter groups with an emphasis on the free objects
Xing Gao, Li Guo, Yanjun Liu +1
Groups with various types of operators, in particular the recently introduced Rota-Baxter groups, have generated renowned interest with close connections to numerical integrals, Ya…
Operator identities on Lie algebras, rewriting systems and Gröbner-Shirshov bases
Huhu Zhang, Xing Gao, Li Guo
Motivated by the pivotal role played by linear operators, many years ago Rota proposed to determine algebraic operator identities satisfied by linear operators on associative algeb…
Rota's program on algebraic operators, rewriting systems and Gröbner-Shirshov bases
Xing Gao, Li Guo, Huhu Zhang
Many years ago, Rota proposed a program on determining algebraic identities that can be satisfied by linear operators. After an extended period of dormant, progress on this program…
Free weighted (modified) differential algebras, free (modified) Rota-Baxter algebras and Gröbner-Shirshov bases
Zhicheng Zhu, Huhu Zhang, Xing Gao
In this paper, we obtain respectively some new linear bases of free unitary (modified) weighted differential algebras and free nonunitary (modified) Rota-Baxter algebras, in terms…
Compatible structures on unary binary nonsymmetric operads with quadratic and cubic relations
Xing Gao, Li Guo, Huhu Zhang
Various compatibility conditions among replicated copies of operations in a given algebraic structure have appeared in broad contexts in recent years. Taking an uniform approach, t…
Commutative matching Rota-Baxter operators, shuffle products with decorations and matching Zinbiel algebras
Xing Gao, Li Guo, Yi Zhang
The Rota-Baxter algebra and the shuffle product are both algebraic structures arising from integral operators and integral equations. Free commutative Rota-Baxter algebras provide…
Operator forms of nonhomogeneous associative classical Yang-Baxter equation
Chengming Bai, Xing Gao, Li Guo +1
This paper studies operator forms of the nonhomogeneous associative classical Yang-Baxter equation (nhacYBe), extending and generalizing such studies for the classical Yang-Baxter…
Free Rota-Baxter family algebras and Free (tri)dendriform family algebras
Yuanyuan Zhang, Xing Gao, Dominique Manchon
In this paper, we first construct the free Rota-Baxter family algebra generated by some set in terms of typed angularly -decorated planar rooted trees. As an application, we…
Hopf algebra of multi-decorated rooted forests, free matching Rota-Baxter algebras and Gröbner-Shirshov bases
Xing Gao, Li Guo, Yi Zhang
Recent advances in stochastic PDEs, Hopf algebras of typed trees and integral equations have inspired the study of algebraic structures with replicating operations. To understand t…
Reynolds algebras and their free objects from bracketed words and rooted trees
Tianjie Zhang, Xing Gao, Li Guo
The study of Reynolds algebras has its origin in the well-known work of O. Reynolds on fluid dynamics in 1895 and has since found broad applications. It also has close relationship…
A note on connected cofiltered coalgebras, conilpotent coalgebras and Hopf algebras
Xing Gao, Li Guo
This notes gives a proof that a connected coaugmented cofiltered coalgebra is a conilpotent coalgebra and thus a connected coaugmented cofiltered bialgebra is a Hopf algebra. This…
Free (tri)dendriform family algebras
Yuanyuan Zhang, Xing Gao, Dominique Manchon
In this paper, we first prove that a Rota-Baxter family algebra indexed by a semigroup induces an ordinary Rota-Baxter algebra structure on the tensor product with the semigroup al…
Matching Rota-Baxter algebras, matching dendriform algebras and matching pre-Lie algebras
Xing Gao, Li Guo, Yi Zhang
We introduce the notion of a matching Rota-Baxter algebra motivated by the recent work on multiple pre-Lie algebras arising from the study of algebraic renormalization of regularit…
Hopf algebras of planar binary trees: an operated algebra approach
Yi Zhang, Xing Gao
Parallel to operated algebras built on top of planar rooted trees via the grafting operator , we introduce and study -algebras and more generally -algebras based…
Free modified Rota-Baxter algebras and Hopf algebras
Xigou Zhang, Xing Gao, Li Guo
The notion of a modified Rota-Baxter algebra comes from the combination of those of a Rota-Baxter algebra and a modified Yang-Baxter equation. In this paper, we first construct fre…
A Hopf algebra on subgraphs of a graph
Xiaomeng Wang, Shoujun Xu, Xing Gao
In this paper, we construct a bialgebraic and further a Hopf algebraic structure on top of subgraphs of a given graph. Further, we give the dual structure of this Hopf algebraic st…
Weighted infinitesimal unitary bialgebras on rooted forests and weighted cocycles
Yi Zhang, Dan Chen, Xing Gao +1
In this paper, we define a new coproduct on the space of decorated planar rooted forests to equip it with a weighted infinitesimal unitary bialgebraic structure. We introduce the c…
Weighted infinitesimal unitary bialgebras on matrix algebras and weighted associative Yang-Baxter equations
Yi Zhang, Xing Gao, Jia-wen Zheng
We equip a matrix algebra with a weighted infinitesimal unitary bialgebraic structure, via a construction of a suitable coproduct. Furthermore, an infinitesimal unitary Hopf algebr…
Hamiltonian decomposition of the Cayley graph on the dihedral group where is a prime
Hui Zhou, Liufeng Xu, Yang Cui +4
In this note, we give the Hamiltonian decomposition of the Cayley graph on the dihedral group where is a prime.
Infinitesimal unitary Hopf algebras and planar rooted forests
Xing Gao, Xiaomeng Wang
Infinitesimal bialgebras were introduced by Joni and Rota. An infinitesimal bialgebra is at the same time an algebra and coalgebra, in such a way that the comultiplication is a der…
Weighted infinitesimal bialgebras
Xing Gao, Yi Zhang
As an algebraic meaning of the nonhomogenous associative Yang-Baxter equation, weighted infinitesimal bialgebras play an important role in mathematics and mathematical physics. In…