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Rota-Baxter Algebra and Related Topics
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Rota-Baxter Algebra and Related Topics

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Algebra, Geometry and Topology".

Deadline for manuscript submissions: closed (30 January 2023) | Viewed by 11231

Special Issue Editors

Dr. Li Guo

E-Mail Website
Guest Editor
Department of Mathematics and Computer Science, Rutgers University, Newark, NJ 08550, USA
Interests: associative and nonassociative algebra and their interplay with combinatorics; number theory and mathematical physics
Dr. Liangyun Zhang

E-Mail Website
Guest Editor
College of Sciences, Nanjing Agricultural University, Nanjing 210095, China
Interests: Hopf algebra; Rota-Baxter algebra; bioinformatics; deep learning

Special Issue Information

Dear Colleagues, 

Rota–Baxter algebra has its origin in the work of G. Baxter, likely around 1960, and was studied by Atkinson, Cartier and Rota in its early years. Independently, Rota–Baxter Lie algebra appeared as the operator form of the classical Yang–Baxter equation. The subject experienced a remarkable renaissance into this century, thanks to its roles in combinatorics, number theory and mathematical physics. Its more recent studies have touched upon even broader areas of research as illustrated below.

You are invited to contribute to the Special Issue on “Rota–Baxter Algebras and Related Topics” of MDPI Mathematics. For further information of the journal, see

https://www.scimagojr.com/journalsearch.php?q=21100830702&tip=sid&clean=0

where the journal, even if very new, is listed in quartile 2 in Mathematics since 2020 with an SJR score of 0.495. This puts the journal at about the same level as Canadian Mathematical Bulletin (0.522), Frontiers of Mathematics in China (0.482), Glasgow Mathematical Journal (0.482) and Bulletin of the Australian Mathematical Society (0.479).

This Special Issue welcomes contributions on Rota–Baxter algebras and related topics. Due to the connections of Rota–Baxter algebras with broad areas in mathematics and mathematical physics, topics covered by the Special Issue include, but are not limited to,

  • Yang–Baxter equations,
  • Algebraic Combinatorics
  • Renormalization issues in physics and mathematics
  • O-operators (aka relative Rota–Baxter operators), and multi-operator structures
  • Multiple zeta values
  • Computational aspects such as Groebner–Shirshov bases
  • Other algebraic operators such as differential, Nijenhuis, averaging and Reynolds operators
  • Rota–Baxter related operators on other structures, such as Hom-structures, groups, Hopf algebras and lattices
  • Representation theoretic aspect
  • Categorical, operadic and universal algebra aspects

Dr. Li Guo
Dr. Liangyun Zhang
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Rota–Baxter operators
  • Yang–Baxter equations
  • differential operators and differential algebras
  • Operads
  • Groebner–Shirshov bases
  • rewriting systems
  • algebraic Combinatorics
  • renormalization
  • multiple zeta values
  • Hopf algebras

Published Papers (7 papers)

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Research

17 pages, 292 KiB  
Article
An Algebraic Characterization of Prefix-Strict Languages
by Jing Tian, Yizhi Chen and Hui Xu
Mathematics 2022, 10(19), 3416; https://doi.org/10.3390/math10193416 - 20 Sep 2022
Viewed by 856
Abstract
Let Σ+ be the set of all finite words over a finite alphabet Σ. A word u is called a strict prefix of a word v, if u is a prefix of v and there is no other way to [...] Read more.
Let Σ+ be the set of all finite words over a finite alphabet Σ. A word u is called a strict prefix of a word v, if u is a prefix of v and there is no other way to show that u is a subword of v. A language L⊆Σ+ is said to be prefix-strict, if for any u,v∈L, u is a subword of v always implies that u is a strict prefix of v. Denote the class of all prefix-strict languages in Σ+ by P(Σ+). This paper characterizes P(Σ+) as a universe of a model of the free object for the ai-semiring variety satisfying the additional identities x+yx≈x and x+yxz≈x. Furthermore, the analogous results for so-called suffix-strict languages and infix-strict languages are introduced. Full article
(This article belongs to the Special Issue Rota-Baxter Algebra and Related Topics)
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12 pages, 322 KiB  
Article
U(h)-Free Modules over the Lie Algebras of Differential Operators
by Munayim Dilxat, Shoulan Gao, Dong Liu and Limeng Xia
Mathematics 2022, 10(10), 1728; https://doi.org/10.3390/math10101728 - 18 May 2022
Viewed by 928
Abstract
This paper mainly considers a class of non-weight modules over the Lie algebra of the Weyl type. First, we construct the U(h)-free modules of rank one over the differential operator algebra. Then, we characterize the tensor products of these [...] Read more.
This paper mainly considers a class of non-weight modules over the Lie algebra of the Weyl type. First, we construct the U(h)-free modules of rank one over the differential operator algebra. Then, we characterize the tensor products of these kind of modules and the quasi-finite highest weight modules. Finally, we undertake such research for the differential operator algebra of multi-variables. Full article
(This article belongs to the Special Issue Rota-Baxter Algebra and Related Topics)
12 pages, 283 KiB  
Article
3-Derivations and 3-Automorphisms on Lie Algebras
by Haobo Xia
Mathematics 2022, 10(5), 782; https://doi.org/10.3390/math10050782 - 28 Feb 2022
Viewed by 1561
Abstract
In this paper, first we establish the explicit relation between 3-derivations and 3- automorphisms of a Lie algebra using the differential and exponential map. More precisely, we show that the Lie algebra of 3-derivations is the Lie algebra of the Lie group of [...] Read more.
In this paper, first we establish the explicit relation between 3-derivations and 3- automorphisms of a Lie algebra using the differential and exponential map. More precisely, we show that the Lie algebra of 3-derivations is the Lie algebra of the Lie group of 3-automorphisms. Then we study the derivations and automorphisms of the standard embedding Lie algebra of a Lie triple system. We prove that derivations and automorphisms of a Lie triple system give rise to derivations and automorphisms of the corresponding standard embedding Lie algebra. Finally we compute the 3-derivations and 3-automorphisms of 3-dimensional real Lie algebras. Full article
(This article belongs to the Special Issue Rota-Baxter Algebra and Related Topics)
20 pages, 334 KiB  
Article
Characterization of Automorphisms of (θ,ω)-Twisted Radford’s Hom-Biproduct
by Xing Wang and Ding-Guo Wang
Mathematics 2022, 10(3), 407; https://doi.org/10.3390/math10030407 - 27 Jan 2022
Viewed by 1651
Abstract
In this paper, we study the Hom–Hopf algebra automorphism group of a (θ,ω)-twisted-Radford’s Hom-biproduct, which satisfies certain conditions. First, we study the endomorphism monoid and automorphism group of (θ,ω)-twisted Radford’s Hom-biproducts, and show [...] Read more.
In this paper, we study the Hom–Hopf algebra automorphism group of a (θ,ω)-twisted-Radford’s Hom-biproduct, which satisfies certain conditions. First, we study the endomorphism monoid and automorphism group of (θ,ω)-twisted Radford’s Hom-biproducts, and show that the endomorphism has a factorization closely related to the factors (A,α) and (H,β). Then, we consider (θ,ω)-twisted Radford’s Hom-biproduct automorphism group AutHom-Hopf(A×ωθH,p) as a subgroup of some semidirect product U(C,A)op⋊φG(A). Finally, we characterize the automorphisms of a concrete example. Full article
(This article belongs to the Special Issue Rota-Baxter Algebra and Related Topics)
15 pages, 288 KiB  
Article
Typed Angularly Decorated Planar Rooted Trees and Ω-Rota-Baxter Algebras
by Yi Zhang, Xiaosong Peng and Yuanyuan Zhang
Mathematics 2022, 10(2), 190; https://doi.org/10.3390/math10020190 - 8 Jan 2022
Viewed by 1763
Abstract
As a generalization of Rota–Baxter algebras, the concept of an Ω-Rota–Baxter could also be regarded as an algebraic abstraction of the integral analysis. In this paper, we introduce the concept of an Ω-dendriform algebra and show the relationship between Ω-Rota–Baxter [...] Read more.
As a generalization of Rota–Baxter algebras, the concept of an Ω-Rota–Baxter could also be regarded as an algebraic abstraction of the integral analysis. In this paper, we introduce the concept of an Ω-dendriform algebra and show the relationship between Ω-Rota–Baxter algebras and Ω-dendriform algebras. Then, we provide a multiplication recursion definition of typed, angularly decorated rooted trees. Finally, we construct the free Ω-Rota–Baxter algebra by typed, angularly decorated rooted trees. Full article
(This article belongs to the Special Issue Rota-Baxter Algebra and Related Topics)
14 pages, 264 KiB  
Article
Rota–Baxter Operators on Cocommutative Weak Hopf Algebras
by Zhongwei Wang, Zhen Guan, Yi Zhang and Liangyun Zhang
Mathematics 2022, 10(1), 95; https://doi.org/10.3390/math10010095 - 28 Dec 2021
Viewed by 1067
Abstract
In this paper, we first introduce the concept of a Rota–Baxter operator on a cocommutative weak Hopf algebra H and give some examples. We then construct Rota–Baxter operators from the normalized integral, antipode, and target map of H. Moreover, we construct a [...] Read more.
In this paper, we first introduce the concept of a Rota–Baxter operator on a cocommutative weak Hopf algebra H and give some examples. We then construct Rota–Baxter operators from the normalized integral, antipode, and target map of H. Moreover, we construct a new multiplication “∗” and an antipode SB from a Rota–Baxter operator B on H such that HB=(H,∗,η,Δ,ε,SB) becomes a new weak Hopf algebra. Finally, all Rota–Baxter operators on a weak Hopf algebra of a matrix algebra are given. Full article
(This article belongs to the Special Issue Rota-Baxter Algebra and Related Topics)
12 pages, 283 KiB  
Article
Limits of Quantum B-Algebras
by Aiping Gan, Aziz Muzammal and Yichuan Yang
Mathematics 2021, 9(24), 3184; https://doi.org/10.3390/math9243184 - 10 Dec 2021
Cited by 1 | Viewed by 2024
Abstract
Every set with a binary operation satisfying a true statement of propositional logic corresponds to a solution of the quantum Yang-Baxter equation. Quantum B-algebras and L-algebras are closely related to Yang-Baxter equation theory. In this paper, we study the categories with [...] Read more.
Every set with a binary operation satisfying a true statement of propositional logic corresponds to a solution of the quantum Yang-Baxter equation. Quantum B-algebras and L-algebras are closely related to Yang-Baxter equation theory. In this paper, we study the categories with quantum B-algebras with morphisms of exact ones or spectral ones. We guarantee the existences of both direct limits and inverse limits. Full article
(This article belongs to the Special Issue Rota-Baxter Algebra and Related Topics)
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