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Journal articles on the topic 'Rota-Baxter operator'

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Published: 26 April 2025

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1

BAI, CHENGMING, LI GUO, and XIANG NI. "RELATIVE ROTA–BAXTER OPERATORS AND TRIDENDRIFORM ALGEBRAS." Journal of Algebra and Its Applications 12, no. 07 (May 16, 2013): 1350027. http://dx.doi.org/10.1142/s0219498813500278.

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A relative Rota–Baxter operator is a relative generalization of a Rota–Baxter operator on an associative algebra. In the Lie algebra context, it is called an [Formula: see text]-operator, originated from the operator form of the classical Yang–Baxter equation. We generalize the well-known construction of dendriform and tridendriform algebras from Rota–Baxter algebras to a construction from relative Rota–Baxter operators. In fact we give two such generalizations, on the domain and range of the operator respectively. We show that each of these generalized constructions recovers all dendriform and tridendriform algebras. Furthermore the construction on the range induces bijections between certain equivalence classes of invertible relative Rota–Baxter operators and tridendriform algebras.

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2

Wang, Zhongwei, Zhen Guan, Yi Zhang, and Liangyun Zhang. "Rota–Baxter Operators on Cocommutative Weak Hopf Algebras." Mathematics 10, no. 1 (December 28, 2021): 95. http://dx.doi.org/10.3390/math10010095.

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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.

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3

Xu, Senrong, Wei Wang, and Jia Zhao. "Twisted Rota-Baxter operators on Hom-Lie algebras." AIMS Mathematics 9, no. 2 (2023): 2619–40. http://dx.doi.org/10.3934/math.2024129.

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<abstract><p>Uchino initiated the investigation of twisted Rota-Baxter operators on associative algebras. Relevant studies have been extensive in recent times. In this paper, we introduce the notion of a twisted Rota-Baxter operator on a Hom-Lie algebra. By utilizing higher derived brackets, we establish an explicit $ L_{\infty} $-algebra whose Maurer-Cartan elements are precisely twisted Rota-Baxter operators on Hom-Lie algebra s. Additionally, we employ Getzler's technique of twisting $ L_\infty $-algebras to establish the cohomology of twisted Rota-Baxter operators. We demonstrate that this cohomology can be regarded as the Chevalley-Eilenberg cohomology of a specific Hom-Lie algebra with coefficients in an appropriate representation. Finally, we study the linear and formal deformations of twisted Rota-Baxter operators by using the cohomology defined above. We also show that the rigidity of a twisted Rota-Baxter operator can be derived from Nijenhuis elements associated with a Hom-Lie algebra.</p></abstract>

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4

Zhao, Jia, and Yu Qiao. "Cohomology and Deformations of Relative Rota–Baxter Operators on Lie-Yamaguti Algebras." Mathematics 12, no. 1 (January 4, 2024): 166. http://dx.doi.org/10.3390/math12010166.

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In this paper, we establish the cohomology of relative Rota–Baxter operators on Lie-Yamaguti algebras via the Yamaguti cohomology. Then, we use this type of cohomology to characterize deformations of relative Rota–Baxter operators on Lie-Yamaguti algebras. We show that if two linear or formal deformations of a relative Rota–Baxter operator are equivalent, then their infinitesimals are in the same cohomology class in the first cohomology group. Moreover, an order n deformation of a relative Rota–Baxter operator can be extended to an order n+1 deformation if and only if the obstruction class in the second cohomology group is trivial.

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5

Guo, Shuangjian, Shengxiang Wang, and Xiaohui Zhang. "The Classical Hom–Leibniz Yang–Baxter Equation and Hom–Leibniz Bialgebras." Mathematics 10, no. 11 (June 3, 2022): 1920. http://dx.doi.org/10.3390/math10111920.

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In this paper, we first introduce the notion of Hom–Leibniz bialgebras, which is equivalent to matched pairs of Hom–Leibniz algebras and Manin triples of Hom–Leibniz algebras. Additionally, we extend the notion of relative Rota–Baxter operators to Hom–Leibniz algebras and prove that there is a Hom–pre-Leibniz algebra structure on Hom–Leibniz algebras that have a relative Rota–Baxter operator. Finally, we study the classical Hom–Leibniz Yang–Baxter equation on Hom–Leibniz algebras and present its connection with the relative Rota–Baxter operator.

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6

Tang, Rong, Yunhe Sheng, and Yanqiu Zhou. "Deformations of relative Rota–Baxter operators on Leibniz algebras." International Journal of Geometric Methods in Modern Physics 17, no. 12 (September 4, 2020): 2050174. http://dx.doi.org/10.1142/s0219887820501741.

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In this paper, we introduce the cohomology theory of relative Rota–Baxter operators on Leibniz algebras. We use the cohomological approach to study linear and formal deformations of relative Rota–Baxter operators. In particular, the notion of Nijenhuis elements is introduced to characterize trivial linear deformations. Formal deformations and extendibility of order [Formula: see text] deformations of a relative Rota–Baxter operator are also characterized in terms of the cohomology theory.

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7

Zhou, Shuyun, and Li Guo. "Rota-Baxter TD Algebra and Quinquedendriform Algebra." Algebra Colloquium 24, no. 01 (February 15, 2017): 53–74. http://dx.doi.org/10.1142/s1005386717000037.

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A dendriform algebra defined by Loday has two binary operations that give a two-part splitting of the associativity in the sense that their sum is associative. Similar dendriform type algebras with three-part and four-part splitting of the associativity were later obtained. These structures can also be derived from actions of suitable linear operators, such as a Rota-Baxter operator or TD operator, on an associative algebra. Motivated by finding a five-part splitting of the associativity, we consider the Rota-Baxter TD (RBTD) operator, an operator combining the Rota-Baxter operator and TD operator, and coming from a recent study of Rota’s problem concerning linear operators on associative algebras. Free RBTD algebras on rooted forests are constructed. We then introduce the concept of a quinquedendriform algebra and show that its defining relations are characterized by the action of an RBTD operator, similar to the cases of dendriform and tridendriform algebras.

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8

Liu, Ling, Abdenacer Makhlouf, Claudia Menini, and Florin Panaite. "-Rota–Baxter Operators, Infinitesimal Hom-bialgebras and the Associative (Bi)Hom-Yang–Baxter Equation." Canadian Mathematical Bulletin 62, no. 02 (January 7, 2019): 355–72. http://dx.doi.org/10.4153/cmb-2018-028-8.

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AbstractWe introduce the concept of a $\{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\}$ -Rota–Baxter operator, as a twisted version of a Rota–Baxter operator of weight zero. We show how to obtain a certain $\{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\}$ -Rota–Baxter operator from a solution of the associative (Bi)Hom-Yang–Baxter equation, and, in a compatible way, a Hom-pre-Lie algebra from an infinitesimal Hom-bialgebra.

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9

Das, Apurba, and Satyendra Kumar Mishra. "The L∞-deformations of associative Rota–Baxter algebras and homotopy Rota–Baxter operators." Journal of Mathematical Physics 63, no. 5 (May 1, 2022): 051703. http://dx.doi.org/10.1063/5.0076566.

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A relative Rota–Baxter algebra is a triple ( A, M, T) consisting of an algebra A, an A-bimodule M, and a relative Rota–Baxter operator T. Using Voronov’s derived bracket and a recent work of Lazarev, Sheng, and Tang, we construct an L∞[1]-algebra whose Maurer–Cartan elements are precisely relative Rota–Baxter algebras. By a standard twisting, we define a new L∞[1]-algebra that controls Maurer–Cartan deformations of a relative Rota–Baxter algebra ( A, M, T). We introduce the cohomology of a relative Rota–Baxter algebra ( A, M, T) and study infinitesimal deformations in terms of this cohomology (in low dimensions). As an application, we deduce cohomology of triangular skew-symmetric infinitesimal bialgebras and discuss their infinitesimal deformations. Finally, we define homotopy relative Rota–Baxter operators and find their relationship with homotopy dendriform algebras and homotopy pre-Lie algebras.

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10

Rosenkranz, Markus, Xing Gao, and Li Guo. "An algebraic study of multivariable integration and linear substitution." Journal of Algebra and Its Applications 18, no. 11 (August 19, 2019): 1950207. http://dx.doi.org/10.1142/s0219498819502074.

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We set up an algebraic theory of multivariable integration, based on a hierarchy of Rota–Baxter operators and an action of the matrix monoid as linear substitutions. Given a suitable coefficient domain with a bialgebra structure, this allows us to build an operator ring that acts naturally on the given Rota–Baxter hierarchy. We conjecture that the operator relations are a noncommutative Gröbner–Shirshov basis for the ideal they generate.

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11

Das, Apurba. "Cohomology and deformations of weighted Rota–Baxter operators." Journal of Mathematical Physics 63, no. 9 (September 1, 2022): 091703. http://dx.doi.org/10.1063/5.0093066.

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Weighted Rota–Baxter operators on associative algebras are closely related to modified Yang–Baxter equations, splitting of algebras, and weighted infinitesimal bialgebras and play an important role in mathematical physics. For any λ ∈ k, we construct a differential graded Lie algebra whose Maurer–Cartan elements are given by λ-weighted relative Rota–Baxter operators. Using such characterization, we define the cohomology of a λ-weighted relative Rota-Baxter operator T and interpret this as the Hochschild cohomology of a suitable algebra with coefficients in an appropriate bimodule. We study linear, formal, and finite order deformations of T from cohomological points of view. Among others, we introduce Nijenhuis elements that generate trivial linear deformations and define a second cohomology class to any finite order deformation, which is the obstruction to extend the deformation. In the end, we also consider the cohomology of λ-weighted relative Rota–Baxter operators in the Lie case and find a connection with the case of associative algebras.

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12

EBRAHIMI-FARD, KURUSCH, and LI GUO. "FREE ROTA–BAXTER ALGEBRAS AND ROOTED TREES." Journal of Algebra and Its Applications 07, no. 02 (April 2008): 167–94. http://dx.doi.org/10.1142/s0219498808002746.

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A Rota–Baxter algebra, also known as a Baxter algebra, is an algebra with a linear operator satisfying a relation, called the Rota–Baxter relation, that generalizes the integration by parts formula. Most of the studies on Rota–Baxter algebras have been for commutative algebras. Two constructions of free commutative Rota–Baxter algebras were obtained by Rota and Cartier in the 1970s and a third one by Keigher and one of the authors in the 1990s in terms of mixable shuffles. Recently, noncommutative Rota–Baxter algebras have appeared both in physics in connection with the work of Connes and Kreimer on renormalization in perturbative quantum field theory, and in mathematics related to the work of Loday and Ronco on dendriform dialgebras and trialgebras. This paper uses rooted trees and forests to give explicit constructions of free noncommutative Rota–Baxter algebras on modules and sets. This highlights the combinatorial nature of Rota–Baxter algebras and facilitates their further study. As an application, we obtain the unitarization of Rota–Baxter algebras.

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13

Burde, Dietrich, and Vsevolod Gubarev. "Decompositions of algebras and post-associative algebra structures." International Journal of Algebra and Computation 30, no. 03 (December 2, 2019): 451–66. http://dx.doi.org/10.1142/s0218196720500071.

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We introduce post-associative algebra structures and study their relationship to post-Lie algebra structures, Rota–Baxter operators and decompositions of associative algebras and Lie algebras. We show several results on the existence of such structures. In particular, we prove that there exists no post-Lie algebra structure on a pair [Formula: see text], where [Formula: see text] is a simple Lie algebra and [Formula: see text] is a reductive Lie algebra, which is not isomorphic to [Formula: see text]. We also show that there is no post-associative algebra structure on a pair [Formula: see text] arising from a Rota–Baxter operator of [Formula: see text], where [Formula: see text] is a semisimple associative algebra and [Formula: see text] is not semisimple. The proofs use results on Rota–Baxter operators and decompositions of algebras.

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14

Liu, Ling, Abdenacer Makhlouf, Claudia Menini, and Florin Panaite. "BiHom-pre-Lie algebras, BiHom-Leibniz algebras and Rota–Baxter operators on BiHom-Lie algebras." Georgian Mathematical Journal 28, no. 4 (July 1, 2021): 581–94. http://dx.doi.org/10.1515/gmj-2021-2094.

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Abstract We contribute to the study of Rota–Baxter operators on types of algebras other than associative and Lie algebras. If A is an algebra of a certain type and R is a Rota–Baxter operator on A, one can define a new multiplication on A by means of R and the previous multiplication and ask under what circumstances the new algebra is of the same type as A. Our first main result deals with such a situation in the case of BiHom-Lie algebras. Our second main result is a BiHom analogue of Aguiar’s theorem that shows how to obtain a pre-Lie algebra from a Rota–Baxter operator of weight zero on a Lie algebra. The BiHom analogue does not work for BiHom-Lie algebras, but for a new concept we introduce here, called left BiHom-Lie algebra, at which we arrived by defining first the BiHom version of Leibniz algebras.

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15

Gubarev, V. "Rota–Baxter operators on a sum of fields." Journal of Algebra and Its Applications 19, no. 06 (July 8, 2019): 2050118. http://dx.doi.org/10.1142/s0219498820501182.

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We count the number of all Rota–Baxter operators (RB-operators) on a finite direct sum [Formula: see text] of fields and count all of them up to conjugation with an automorphism. We also study RB-operators on [Formula: see text] corresponding to a decomposition of [Formula: see text] into a direct vector space sum of two subalgebras. We show that every algebra structure induced on [Formula: see text] by a RB-operator of nonzero weight is isomorphic to [Formula: see text].

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16

GUO, LI, and ZHONGKUI LIU. "ROTA–BAXTER OPERATORS ON GENERALIZED POWER SERIES RINGS." Journal of Algebra and Its Applications 08, no. 04 (August 2009): 557–64. http://dx.doi.org/10.1142/s0219498809003515.

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An important instance of Rota–Baxter algebras from their quantum field theory application is the ring of Laurent series with a suitable projection. We view the ring of Laurent series as a special case of generalized power series rings with exponents in an ordered monoid. We study when a generalized power series ring has a Rota–Baxter operator and how this is related to the ordered monoid.

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17

Gao, Xing, Li Guo, and Markus Rosenkranz. "On rings of differential Rota–Baxter operators." International Journal of Algebra and Computation 28, no. 01 (February 2018): 1–36. http://dx.doi.org/10.1142/s0218196718500017.

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Using the language of operated algebras, we construct and investigate a class of operator rings and enriched modules induced by a derivation or Rota–Baxter operator. In applying the general framework to univariate polynomials, one is led to the integro–differential analogs of the classical Weyl algebra. These are analyzed in terms of skew polynomial rings and noncommutative Gröbner bases.

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18

Mazurek, Ryszard. "Rota–Baxter operators on skew generalized power series rings." Journal of Algebra and Its Applications 13, no. 07 (May 2, 2014): 1450048. http://dx.doi.org/10.1142/s0219498814500480.

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Let R be a ring, S a strictly ordered monoid, and ω : S → End (R) a monoid homomorphism. The skew generalized power series ring R[[S, ω]] is a common generalization of (skew) polynomial rings, (skew) Laurent polynomial rings, (skew) power series rings, (skew) Laurent series rings, (skew) monoid rings, (skew) Mal'cev–Neumann series rings, and generalized power series rings. We characterize those subsets T of S for which the cut-off operator with respect to T is a Rota–Baxter operator on the ring R[[S, ω]]. The obtained results provide a large class of noncommutative Rota–Baxter algebras.

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19

Hou, Dongping, Xiang Ni, and Chengming Bai. "Pre-jordan Algebras." MATHEMATICA SCANDINAVICA 112, no. 1 (March 1, 2013): 19. http://dx.doi.org/10.7146/math.scand.a-15231.

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The purpose of this paper is to introduce and study a notion of pre-Jordan algebra. Pre-Jordan algebras are regarded as the underlying algebraic structures of the Jordan algebras with a nondegenerate symplectic form. They are the algebraic structures behind the Jordan Yang-Baxter equation and Rota-Baxter operators in terms of $\mathcal{O}$-operators of Jordan algebras introduced in this paper. Pre-Jordan algebras are analogues for Jordan algebras of pre-Lie algebras and fit into a bigger framework with a close relationship with dendriform algebras. The anticommutator of a pre-Jordan algebra is a Jordan algebra and the left multiplication operators give a representation of the Jordan algebra, which is the beauty of such a structure. Furthermore, we introduce a notion of $\mathcal{O}$-operator of a pre-Jordan algebra which gives an analogue of the classical Yang-Baxter equation in a pre-Jordan algebra.

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20

史, 新颖. "Relative Rota-Baxter Operator of Non-Zero Weights on the Jordan Triple." Advances in Applied Mathematics 12, no. 05 (2023): 2468–79. http://dx.doi.org/10.12677/aam.2023.125249.

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21

Martinez, Wilson Arley, Enrique G. Reyes, and Maria Ronco. "Generalizing dendriform algebras: Dyckm-algebras, Rotam-algebras, and Rota–Baxter operators." International Journal of Geometric Methods in Modern Physics 18, no. 11 (August 5, 2021): 2150176. http://dx.doi.org/10.1142/s0219887821501760.

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We review the notion of a [Formula: see text], an algebraic structure introduced recently by López, Préville-Ratelle and Ronco during their work on the splitting of associativity via [Formula: see text]-Dyck paths, and we also introduce Rota[Formula: see text]-algebras: both structures can be considered as generalizations of dendriform structures. We obtain examples of Dyck[Formula: see text]-algebras in terms of planar rooted binary trees equipped with a particular type of Rota–Baxter operator, and we present examples of Rotam-algebras using left averaging morphisms. As an application, we observe that the structures presented here allow us to introduce quite naturally a “non-associative version” of the Kadomtsev–Petviashvili hierarchy.

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22

Pietrzkowski, Gabriel. "Inhomogeneous Spitzer-type identities for commutative and non-commutative Rota–Baxter algebras." Journal of Algebra and Its Applications 16, no. 01 (January 2017): 1750016. http://dx.doi.org/10.1142/s0219498817500165.

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We consider a complete filtered Rota–Baxter (RB) algebra of weight [Formula: see text] over a commutative ring. Finding the unique solution of an inhomogeneous linear algebraic equation in this algebra, we generalize Spitzer’s identity in both commutative and non-commutative cases. As an application, considering the RB algebra of power series in one variable with q-integral as the RB operator, we show certain Eulerian identities.

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23

Gustavson, Richard, and Sarah Rosen. "A Reduction Algorithm for Volterra Integral Equations." PUMP Journal of Undergraduate Research 6 (May 30, 2023): 172–91. http://dx.doi.org/10.46787/pump.v6i0.3631.

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An integral equation is a way to encapsulate the relationships between a function and its integrals. We develop a systematic way of describing Volterra integral equations – specifically an algorithm that reduces any separable Volterra integral equation into an equivalent one in operator-linear form, i.e., one that only contains iterated integrals. This serves to standardize the presentation of such integral equations so as to only consider those containing iterated integrals. We use the algebraic object of the integral operator, the twisted Rota-Baxter identity, and vertex-edge decorated rooted trees to construct our algorithm.

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24

Zhang, Shilong, Li Guo, and William Keigher. "Classification of operator extensions, monad liftings and distributive laws for differential algebras and Rota–Baxter algebras." Journal of Algebra and Its Applications 19, no. 09 (September 19, 2019): 2050172. http://dx.doi.org/10.1142/s0219498820501728.

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Generalizing the algebraic formulation of the First Fundamental Theorem of Calculus (FFTC), a class of constraints involving a pair of operators was considered in [Extensions of operators, liftings of monads, and mixed distributive laws, Appl. Categ. Struct. 26 (2018) 747–765]. For a given constraint, the existences of extensions of differential and Rota–Baxter operators, of liftings of monads and comonads, and of mixed distributive laws are shown to be equivalent. In this paper, we give a classification of the constraints satisfying these equivalent conditions.

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25

Cao, X., S. H. Liu, X. S. Lu, Z. J. Ye, Z. R. Yu, and Y. H. Zhang. "Derimorphisms over Algebras and Applications." Algebra Colloquium 30, no. 02 (June 2023): 193–204. http://dx.doi.org/10.1142/s1005386723000160.

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The new concept “derimorphism” generalizing both derivation and homomorphism is defined. When a derimorphism is invertible, its inverse is a Rota–Baxter operator. The general theory of derimorphism is established. The classification of all derimorphisms over an associative unital algebra is obtained. Contrary to the nonexistence of nontrivial positive derivations, it is shown that nontrivial positive derimorphisms do exist over any pair of opposite orderings over [Formula: see text], the lattice-ordered full matrix algebra and upper triangular matrix algebra over a totally ordered field.

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Chtioui, T., S. Mabrouk, and A. Makhlouf. "Construction of Hom-pre-Jordan algebras and Hom-J-dendriform algebras." Extracta Mathematicae 38, no. 1 (June 1, 2023): 27–50. http://dx.doi.org/10.17398/2605-5686.38.1.27.

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The aim of this work is to introduce and study the notions of Hom-pre-Jordan algebra and Hom-J-dendriform algebra which generalize Hom-Jordan algebras. Hom-pre-Jordan algebras are regarded as the underlying algebraic structures of the Hom-Jordan algebras behind the Rota-Baxter operators and O-operators introduced in this paper. Hom-pre-Jordan algebras are also analogues of Hom-pre-Lie algebras for Hom-Jordan algebras. The anti-commutator of a Hom-pre-Jordan algebra is a Hom-Jordan algebra and the left multiplication operator gives a representation of a Hom-Jordan algebra. On the other hand, a Hom-J-dendriform algebra is a Hom-Jordan algebraic analogue of a Hom-dendriform algebra such that the anti-commutator of the sum of the two operations is a Hom-pre-Jordan algebra.

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Wang, Qi, Yunhe Sheng, Chengming Bai, and Jiefeng Liu. "Nijenhuis operators on pre-Lie algebras." Communications in Contemporary Mathematics 21, no. 07 (October 10, 2019): 1850050. http://dx.doi.org/10.1142/s0219199718500505.

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First we use a new approach to define a graded Lie algebra whose Maurer–Cartan elements characterize pre-Lie algebra structures. Then using this graded Lie bracket, we define the notion of a Nijenhuis operator on a pre-Lie algebra which generates a trivial deformation of this pre-Lie algebra. There are close relationships between [Formula: see text]-operators, Rota–Baxter operators and Nijenhuis operators on a pre-Lie algebra. In particular, a Nijenhuis operator “connects” two [Formula: see text]-operators on a pre-Lie algebra whose any linear combination is still an [Formula: see text]-operator in certain sense and hence compatible [Formula: see text]-dendriform algebras appear naturally as the induced algebraic structures. For the case of the dual representation of the regular representation of a pre-Lie algebra, there is a geometric interpretation by introducing the notion of a pseudo-Hessian–Nijenhuis structure which gives rise to a sequence of pseudo-Hessian and pseudo-Hessian–Nijenhuis structures. Another application of Nijenhuis operators on pre-Lie algebras in geometry is illustrated by introducing the notion of a para-complex structure on a pre-Lie algebra and then studying para-complex quadratic pre-Lie algebras and para-complex pseudo-Hessian pre-Lie algebras in detail. Finally, we give some examples of Nijenhuis operators on pre-Lie algebras.

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28

Aloulou, Walid, and Mansour Jebli. "Structure of BiHom-pre-Poisson algebras." Acta et Commentationes Universitatis Tartuensis de Mathematica 28, no. 2 (November 28, 2024): 149–73. http://dx.doi.org/10.12697/acutm.2024.28.11.

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In the current research paper, we define and investigate the structure of a BiHom-pre-Poisson algebra. This algebraic structure is defined by two products "Λ", "◊" and two linear maps f, g on A. In particular, (A, Λ, f, g) is a BiHom-Zinbiel algebra and (A, ◊, f, g) is a BiHom-pre-Lie algebra. Additionally two compatibility conditions between Λ and ◊ are verified. Our first main results are devoted to demonstrating that if A is a BiHom-pre-Lie algebra, then a tensorial algebra of A has a structure of a BiHom-pre-Poisson algebra. Furthermore, we prove that any BiHom-Poisson algebra together with a Rota–Baxter operator defines a BiHom-pre-Poisson algebra. Finally, we define the structure of a dual BiHom-pre-Poisson algebra and we demonstrate that an averaging operator on a BiHom-Poisson algebra gives rise to a dual BiHom-pre-Poisson algebra.

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29

Wang, Ximu, Chongxia Zhang, and Liangyun Zhang. "Rota–Baxter Operators on Skew Braces." Mathematics 12, no. 11 (May 27, 2024): 1671. http://dx.doi.org/10.3390/math12111671.

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In this paper, we introduce the concept of Rota–Baxter skew braces, and provide classifications of Rota–Baxter operators on various skew braces, such as (Z,+,∘) and (Z/(4),+,∘). We also present a necessary and sufficient condition for a skew brace to be a co-inverse skew brace. Additionally, we describe some constructions of Rota–Baxter quasiskew braces, and demonstrate that every Rota–Baxter skew brace can induce a quasigroup and a Rota–Baxter quasiskew brace.

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30

Gubarev, Vsevolod. "Rota-Baxter operators and Bernoulli polynomials." Communications in Mathematics 29, no. 1 (April 30, 2021): 1–14. http://dx.doi.org/10.2478/cm-2021-0001.

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Abstract We develop the connection between Rota-Baxter operators arisen from algebra and mathematical physics and Bernoulli polynomials. We state that a trivial property of Rota-Baxter operators implies the symmetry of the power sum polynomials and Bernoulli polynomials. We show how Rota-Baxter operators equalities rewritten in terms of Bernoulli polynomials generate identities for the latter.

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31

Azizov, M. E. "Anti-Rota-Baxter operators on Witt and Virasoro algebras." UZBEK MATHEMATICAL JOURNAL 68, no. 4 (February 13, 2025): 26–36. https://doi.org/10.29229/uzmj.2024-4-3.

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In this work, we obtain the description of all homogeneous anti-Rota-Baxter operators on Witt and Virasoro algebras. Moreover, we describe anti-Rota-Baxter operators on three-dimensional simple Lie algebra sl2

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32

Zhang, Liangyun, Linhan Li, and Huihui Zheng. "Rota-Baxter Leibniz Algebras and Their Constructions." Advances in Mathematical Physics 2018 (December 2, 2018): 1–15. http://dx.doi.org/10.1155/2018/8540674.

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In this paper, we introduce the concept of Rota-Baxter Leibniz algebras and explore two characterizations of Rota-Baxter Leibniz algebras. And we construct a number of Rota-Baxter Leibniz algebras from Leibniz algebras and associative algebras and discover some Rota-Baxter Leibniz algebras from augmented algebra, bialgebra, and weak Hopf algebra. In the end, we give all Rota-Baxter operators of weight 0 and -1 on solvable and nilpotent Leibniz algebras of dimension ≤3, respectively.

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33

QIU, JIANJUN, and YUQUN CHEN. "COMPOSITION-DIAMOND LEMMA FOR λ-DIFFERENTIAL ASSOCIATIVE ALGEBRAS WITH MULTIPLE OPERATORS." Journal of Algebra and Its Applications 09, no. 02 (April 2010): 223–39. http://dx.doi.org/10.1142/s0219498810003859.

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In this paper, we establish the Composition-Diamond lemma for λ-differential associative algebras over a field K with multiple operators. As applications, we obtain Gröbner–Shirshov bases of free λ-differential Rota–Baxter algebras. In particular, linear bases of free λ-differential Rota–Baxter algebras are obtained and consequently, the free λ-differential Rota–Baxter algebras are constructed by words.

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34

Wu, Linli, Mengping Wang, and Yongsheng Cheng. "Rota-Baxter Operators on 3-Dimensional Lie Algebras and the Classical R-Matrices." Advances in Mathematical Physics 2017 (2017): 1–7. http://dx.doi.org/10.1155/2017/6128102.

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Our aim is to classify the Rota-Baxter operators of weight 0 on the 3-dimensional Lie algebra whose derived algebra’s dimension is 2. We explicitly determine all Rota-Baxter operators (of weight zero) on the 3-dimensional Lie algebras g. Furthermore, we give the corresponding solutions of the classical Yang-Baxter equation in the 6-dimensional Lie algebras g ⋉ad⁎ g⁎ and the induced left-symmetry algebra structures on g.

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35

Yu, Houyi. "Classification of monomial Rota–Baxter operators on k[x]." Journal of Algebra and Its Applications 15, no. 05 (March 30, 2016): 1650087. http://dx.doi.org/10.1142/s0219498816500870.

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Rota–Baxter operators were introduced to solve certain analytic and combinatorial problems and then applied to many fields in mathematics and mathematical physics. The polynomial algebra [Formula: see text] plays a central role both in analysis and algebra. In this paper, we explicitly classified all monomial Rota–Baxter operators on [Formula: see text].

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36

Gubarev, V. "Rota–Baxter Operators on Unital Algebras." Moscow Mathematical Journal 21, no. 2 (2021): 325–64. http://dx.doi.org/10.17323/1609-4514-2021-21-2-325-364.

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37

Das, Apurba. "Deformations of associative Rota-Baxter operators." Journal of Algebra 560 (October 2020): 144–80. http://dx.doi.org/10.1016/j.jalgebra.2020.05.016.

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38

龚, 晓倩. "Rota-Baxter Operators on Clifford Semigroups." Pure Mathematics 14, no. 05 (2024): 590–98. http://dx.doi.org/10.12677/pm.2024.145212.

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39

Qiu, Jianjun. "Gröbner–Shirshov bases for commutative algebras with multiple operators and free commutative Rota–Baxter algebras." Asian-European Journal of Mathematics 07, no. 02 (June 2014): 1450033. http://dx.doi.org/10.1142/s1793557114500338.

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In this paper, the Composition-Diamond lemma for commutative algebras with multiple operators is established. As applications, the Gröbner–Shirshov bases and linear bases of free commutative Rota–Baxter algebra, free commutative λ-differential algebra and free commutative λ-differential Rota–Baxter algebra are given, respectively. Consequently, these three free algebras are constructed directly by commutative Ω-words.

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40

Abdujabborov, N. G., I. A. Karimjanov, and M. A. Kodirova. "Rota-type operators on 3-dimensional nilpotent associative algebras." Communications in Mathematics 29, no. 2 (June 1, 2021): 227–41. http://dx.doi.org/10.2478/cm-2021-0020.

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41

Goncharov, Maxim. "Rota-Baxter operators on cocommutative Hopf algebras." Journal of Algebra 582 (September 2021): 39–56. http://dx.doi.org/10.1016/j.jalgebra.2021.04.024.

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42

Das, Apurba. "Rota--Baxter operators on involutive associative algebras." Acta Scientiarum Mathematicarum 87, no. 34 (2021): 349–66. http://dx.doi.org/10.14232/actasm-020-616-0.

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43

Li, Xiuxian, Dongping Hou, and Chengming Bai. "Rota-Baxter operators on pre-Lie algebras." Journal of Nonlinear Mathematical Physics 14, no. 2 (January 2007): 269–89. http://dx.doi.org/10.2991/jnmp.2007.14.2.10.

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44

Panaite, Florin, and Freddy Van Oystaeyen. "Twisted algebras and Rota–Baxter type operators." Journal of Algebra and Its Applications 16, no. 04 (April 2017): 1750079. http://dx.doi.org/10.1142/s0219498817500797.

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We define the concept of weak pseudotwistor for an algebra [Formula: see text] in a monoidal category [Formula: see text], as a morphism [Formula: see text] in [Formula: see text], satisfying some axioms ensuring that [Formula: see text] is also an algebra in [Formula: see text]. This concept generalizes the previous proposal called pseudotwistor and covers a number of examples of twisted algebras that cannot be covered by pseudotwistors, mainly examples provided by Rota–Baxter operators and some of their relatives (such as Leroux’s TD-operators and Reynolds operators). By using weak pseudotwistors, we introduce an equivalence relation (called “twist equivalence”) for algebras in a given monoidal category.

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45

Abdaoui, El-Kadri, Sami Mabrouk, and Abdenacer Makhlouf. "Rota–Baxter Operators on Pre-Lie Superalgebras." Bulletin of the Malaysian Mathematical Sciences Society 42, no. 4 (December 26, 2017): 1567–606. http://dx.doi.org/10.1007/s40840-017-0565-x.

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王, 智斌. "Rota-Baxter Operators on Two-Dimensional Pre-Jordan Algebras." Pure Mathematics 13, no. 11 (2023): 3154–64. http://dx.doi.org/10.12677/pm.2023.1311327.

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47

Gao, Xu, Ming Liu, Chengming Bai, and Naihuan Jing. "Rota–Baxter operators on Witt and Virasoro algebras." Journal of Geometry and Physics 108 (October 2016): 1–20. http://dx.doi.org/10.1016/j.geomphys.2016.06.007.

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Dassoundo, Mafoya Landry. "Relative (pre-)anti-flexible algebrasnand associated algebraic structures." Quasigroups and Related Systems 30, no. 1(47) (May 2022): 31–46. http://dx.doi.org/10.56415/qrs.v30.03.

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Pre-anti-flexible family algebras are introduced and used to define and describe the notions of Ωc-relative anti-flexible algebras, left and right pre-Lie family algebras and Ωc-relative Lie algebras. The notion of Ωc-relative pre-anti-flexible algebras are introduced and also used to characterize pre-anti-flexible family algebras, left and right pre-Lie family algebras and significant identities associated to these algebraic structures are provided. Finally, a generalization of the Rota-Baxter operators defined on an Ωc-relative anti-flexible algebra is introduced and it is also proved that both Rota-Baxter operators and its generalization provide Ωc-relative pre-antiflexible algebras structures and related consequences are derived.

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Gubarev, Vsevolod. "Embedding of Pre-Lie Algebras into Preassociative Algebras." Algebra Colloquium 27, no. 02 (May 7, 2020): 299–310. http://dx.doi.org/10.1142/s1005386720000243.

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Zheng, Shanghua, Li Guo, and Markus Rosenkranz. "Rota–Baxter operators on the polynomial algebra, integration, and averaging operators." Pacific Journal of Mathematics 275, no. 2 (May 15, 2015): 481–507. http://dx.doi.org/10.2140/pjm.2015.275.481.

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