Li Guo, William Keigher
In this paper we generalize the well-known construction of shuffle product algebras by using mixable shuffles, and prove that any free Baxter algebra is isomorphic to a mixable shuffle product algebra. This gives an explicit construction of the free Baxter algebra, extending the work of Rota and Cartier.
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We study the ring theoretical structures of mixable shuffle algebras and their associated free commutative Rota-Baxter algebras. For this study we utilize the connection of the mixable shuffle algebras with the overlapping shuffle algebra of Hazewinkel, quasi-shuffle algebras of Hoffman and quasi-symmetric functions. This connection allows us to apply methods and results on shuffle products and Lyndon words on ordered sets. As a result, we obtain structure theorems for a large class of mixable shuffle algebras and free commutative Rota-Baxter algebras with various coefficient rings.
In this paper, we begin a systematic study of modified Rota-Baxter algebras, as an associative analogue of the modified classical Yang-Baxter equation. We construct free commutative modified Rota-Baxter algebras by a variation of the shuffle product and describe the structure both recursively and explicitly. We then provide these algebras with a Hopf algebra structure by applying a Hochschild cocycle.
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 an algebraic framework for integral equations with the simple Riemann integral operator. The Zinbiel algebras form a category in which the shuffle product algebra is the free object. Motivated by algebraic structures underlying integral equations involving multiple integral operators and kernels, we study commutative matching Rota-Baxter algebras and construct the free objects making use of the shuffle product with m…
We show that the shuffle algebras for polylogarithms and regularized MZVs in the sense of Ihara, Kaneko and Zagier are both free commutative nonunitary Rota-Baxter algebras with one generator. We apply these results to show that the full sets of shuffle relations of polylogarithms and regularized MZVs are derived by a single series. We also take this approach to study the extended double shuffle relations of MZVs by comparing these shuffle relations with the quasi-shuffle relations of the regularized MZVs in our previous approach of MZVs by renormalization.
In this letter, we use quantum quasi-shuffle algebras to construct Rota-Baxter algebras, as well as tridendriform algebras. We also propose the notion of braided Rota-Baxter algebras, which is the relevant object of Rota-Baxter algebras in a braided tensor category. Examples of such new algebras are provided by using quantum multi-brace algebras in a category of Yetter-Drinfeld modules.
Rota-Baxter algebras and the closely related dendriform algebras have important physics applications, especially to renormalization of quantum field theory. Braided structures provide effective ways of quantization such as for quantum groups. Continuing recent study relating the two structures, this paper considers Rota-Baxter algebras and dendriform algebras in the braided contexts. Applying the quantum shuffle and quantum quasi-shuffle products, we construct free objects in the categories of braided Rota-Baxter algebras and braided dendriform algebras, under the commutativity condition. We f…
We continue the study of free Baxter algebras. There are two goals of this paper. The first goal is to extend the construction of shuffle Baxter algebras to completions of Baxter algebras. This process is motivated by a construction of Cartier and is analogous to the process of completing a polynomial algebra to obtain a power series algebra. However, as we will see later, unlike the close similarity of properties of a polynomial algebra and a power series algebra, properties of a shuffle Baxter algebra and its completion can be quite different.
Shuffle and quasi-shuffle products are well-known in the mathematics literature. They are intimately related to Loday's dendriform algebras, and were extensively used to give explicit constructions of free commutative Rota-Baxter algebras. In the literature there exist at least two other Rota-Baxter type algebras, namely, the Nijenhuis algebra and the so-called TD-algebra. The explicit construction of the free unital commutative Nijenhuis algebra uses a modified quasi-shuffle product, called the right-shift shuffle. We show that another modification of the quasi-shuffle product, the so-called …
The quasi-shuffle product and mixable shuffle product are both generalizations of the shuffle product and have both been studied quite extensively recently. We relate these two generalizations and realize quasi-shuffle product algebras as subalgebras of mixable shuffle product algebras. As an application, we obtain Hopf algebra structures in free Rota-Baxter algebras.
A Rota-Baxter operator of weight $λ$ is an abstraction of both the integral operator (when $λ=0$) and the summation operator (when $λ=1$). We similarly define a differential operator of weight $λ$ that includes both the differential operator (when $λ=0$) and the difference operator (when $λ=1$). We further consider an algebraic structure with both a differential operator of weight $λ$ and a Rota-Baxter operator of weight $λ$ that are related in the same way that the differential operator and the integral operator are related by the First Fundamental Theorem of Calculus. We construct free objec…
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 Krei…
In recent years, the usual BPHZ algorithm for renormalization in perturbative quantum field theory has been interpreted, after dimensional regularization, as a Birkhoff decomposition of characters on the Hopf algebra of Feynman graphs, with values in a Rota-Baxter algebra of amplitudes. We associate in this paper to any such algebra a universal semi-group (different in nature from the Connes-Marcolli "cosmical Galois group"). Its action on the physical amplitudes associated to Feynman graphs produces the expected operations: Bogoliubov's preparation map, extraction of divergences, renormalizat…
The theory of operated algebras has played a pivotal role in mathematics and physics. In this paper, we introduce a $λ$-TD algebra that appropriately includes both the Rota-Baxter algebra and the TD-algebra. The explicit construction of free commutative $λ$-TD algebra on a commutative algebra is obtained by generalized shuffle products, called $λ$-TD shuffle products. We then show that the free commutative $λ$-TD algebra possesses a left counital bialgera structure by means of a suitable 1-cocycle condition. Furthermore, the classical result that every connected filtered bialgebra is a Hopf al…
The Hopf algebra of word-quasi-symmetric functions ($\WQSym$), a noncommutative generalization of the Hopf algebra of quasi-symmetric functions, can be endowed with an internal product that has several compatibility properties with the other operations on $\WQSym$. This extends constructions familiar and central in the theory of free Lie algebras, noncommutative symmetric functions and their various applications fields, and allows to interpret $\WQSym$ as a convolution algebra of linear endomorphisms of quasi-shuffle algebras. We then use this interpretation to study the fine structure of quas…
Using the concept of mixable shuffles, we formulate explicitly the quantum quasi-shuffle product, as well as the subalgebra generated by primitive elements of the quantum quasi-shuffle bialgebra. We construct a braided coalgebra structure which is dual to the quantum quasi-shuffle algebra. We provide representations of quantum quasi-shuffle algebras on commutative braided Rota-Baxter algebras. As an application, we establish a formal power series whose terms come from a special representation of some kind of quasi-shuffle algebra and whose evaluation at 1 is the multiple $q$-zeta values.
Many features of classical Lie theory generalize to the broader context of algebras over Hopf operads. However, this idea remains largely to be developed systematically. Quasi-shuffle algebras provide for example an interesting illustration of these phenomena, but have not been investigated from this point of view.The notion of quasi-shuffle algebras can be traced back to the beginings of the theory of Rota--Baxter algebras, but was developed systematically only recently, starting essentially with Hoffman's work, that was motivated by multizeta values (MZVs) and featured their bialgebra struct…
From iterated integrals and chronological calculus to Hopf and Rota-Baxter algebras
本综述统一呈现迭代积分与时序演算通向Hopf与Rota-Baxter代数的三条发展路线，提出几何与代数两种视角，涵盖非交换Rota-Baxter代数及相关结构。
Extended Shuffle Product for Multiple Zeta Values
本文通过用微分算子替换积分算子，构造扩展洗牌代数以编码任意整数参数的收敛多重 zeta 级数，并证明相应子空间构成子代数且求和映射为代数同态。
We introduce in this paper the generalized virtual braid group on n strands GVB_n, generalizing simultaneously the braid groups and their virtual versions. A Mastumoto-Tits type section lifting shuffles in a symmetric group S_n to the monoid associated to GVB_n is constructed, which is then applied to characterize the quantum quasi-shuffle product. A family of representations of GVB_n is constructed using quantum groups.
In this survey, we outline two recent constructions of free commutative integro-differential algebras. They are based on the construction of free commutative Rota-Baxter algebras by mixable shuffles. The first is by evaluations. The second is by the method of Gröbner-Shirshov bases.
In this note we introduce the concept of a shuffle product $\sq$ for planar tree polynomials and give a formula to compute the planar shuffle product $S \ \sq T$ of two finite planar reduced rooted trees $S, T.$ It is shown that $\sq$ is dual to the co-addition $Δ$ which leads to a formula for the coefficients of $Δ(f).$ It is also proved that $Δ(EXP) = EXP {\hat \otimes} EXP$ where $EXP$ is the generic planar tree exponential series, see [G]. Systems of quadratic relations for the coefficients of EXP are derived.
Extended Rota-Baxter algebras, diagonally colored Delannoy paths and Hopf algebras
本文系统研究扩张 Rota-Baxter 算子，通过推广拟洗牌乘积构造自由交换对象，并用着色 Delannoy 路径解释其乘法，最终建立 Hopf 代数结构。
Generalized Reynolds algebras from Volterra integrals and their free construction by complete shuffle product
本文引入Volterra积分算子的广义Reynolds代数结构，通过完全shuffle积构造自由对象，以代数方法统一研究相关积分方程。
This paper provides a general operadic definition for the notion of splitting the operations of algebraic structures. This construction is proved to be equivalent to some Manin products of operads and it is shown to be closely related to Rota-Baxter operators. Hence, it gives a new effective way to compute Manin black products. The present construction is shown to have symmetry properties. Finally, this allows us to describe the algebraic structure of square matrices with coefficients in algebras of certain types. Many examples illustrate this text, including the case of Jordan algebras.
A commutative Rota-Baxter algebra can be regarded as a commutative algebra that carries an abstraction of the integral operator. With the motivation of generalizing the study of algebraic geometry to Rota-Baxter algebra, we extend the central concept of localization for commutative algebras to commutative Rota-Baxter algebras. The existence of such a localization is proved and, under mild conditions, its explicit constructions are obtained. The existence of tensor products of commutative Rota-Baxter algebras is also proved and the compatibility of localization and tensor product of Rota-Baxter…
Constructions of Rota-Baxter operators by L-R smash products
本文在L-R smash积上刻画Rota-Baxter算子并给出构造条件，同时在对偶的L-R smash余积上构造Rota-Baxter余算子。
Rota-Baxter operators are an algebraic abstraction of integration. Following this classical connection, we study the relationship between Rota-Baxter operators and integrals in the case of the polynomial algebra $\mathbf{k}[x]$. We consider two classes of Rota-Baxter operators, monomial ones and injective ones. For the first class, we apply averaging operators to determine monomial Rota-Baxter operators. For the second class, we make use of the double product on Rota-Baxter algebras.
Commutative shuffle products are known to be intimately related to universal formulas for products, exponentials and logarithms in group theory as well as in the theory of free Lie algebras, such as, for instance, the Baker-Campbell-Hausdorff formula or the analytic expression of a Lie group law in exponential coordinates in the neighbourhood of the identity. Non-commutative shuffle products happen to have similar properties with respect to pre-Lie algebras. However, the situation is more complex since in the non-commutative framework three exponential-type maps and corresponding logarithms ar…
We introduce the concept of an extended O-operator that generalizes the well-known concept of a Rota-Baxter operator. We study the associative products coming from these operators and establish the relationship between extended O-operators and the associative Yang-Baxter equation, extended associative Yang-Baxter equation and generalized Yang-Baxter equation.
We introduce the notion of quadri-algebras. These are associative algebras for which the multiplication can be decomposed as the sum of four operations in a certain coherent manner. We present several examples of quadri-algebras: the algebra of permutations, the shuffle algebra, tensor products of dendriform algebras. We show that a pair of commuting Baxter operators on an associative algebra gives rise to a canonical quadri-algebra structure on the underlying space of the algebra. The main example is provided by the algebra End(A) of linear endomorphisms of an infinitesimal bialgebra A. This …
Segre products and Segre morphisms in a class of Yang-Baxter algebras
本文研究了 Yang-Baxter 方程的有限非退化对合集合论解所对应的二次 Yang-Baxter 代数，显式给出了 Segre 积的单生成子与二次关系表示，引入了 Segre 映射的类似物并求出其像与核，且结果与经典交换情形一致。
Skew braces and Rota-Baxter operators on semi-direct products
本文研究了斜左 brace 与半直积上的 Rota-Baxter 运算符关系，定义了斜左 brace 的平方结构，提供了一种构造 Yang-Baxter 方程新解的方法，并证明了该构造的泛性质。
Double cross products with projections and relative Rota-Baxter operators on Hopf algebras
本文研究带投影的双交叉积，证明其等价于Hopf代数上的相对Rota-Baxter算子，并探讨其与辫子算子及Doi-Hopf模态射的联系。
This paper studies formal deformations and homotopy theory of Rota-Baxter algebras of any weight. We define an $L_\infty$-algebra, which controls simultaneous deformations of associative products and Rota-Baxter operators. As a consequence, we develop a cohomology theory of Rota-Baxter algebras of any weight and justify it by interpreting lower degree cohomology groups as formal deformations and abelian extensions. The notion of homotopy Rota-Baxter algebras is introduced and it is shown that the operad governing homotopy Rota-Baxter algebras is a minimal model of the operad of Rota-Baxter alg…
Rota-Baxter systems are modified by the inclusion of a curvature term. It is shown that, subject to specific properties of the curvature form, curved Rota-Baxter systems $(A,R,S,ω)$ induce associative and (left) pre-Lie products on the algebra $A$. It is also shown that if both Rota-Baxter operators coincide with each other and the curvature is $A$-bilinear, then the (modified by $R$) Hochschild cohomology ring over $A$ is a curved differential graded algebra.
Multiple Rota-Baxter algebra and multiple Rota-Baxter modules
本文发展了多重Rota-Baxter代数上的模理论，引入左右双模结构，构造自由模，给出充要条件，定义投射内射模并证明范畴有足够对象，最后定义张量积和平坦模。
This paper shows that every finite non-degenerate involutive set theoretic solution (X,r) of the Yang-Baxter equation whose symmetric group has cardinality which a cube-free number is a multipermutation solution. Some properties of finite braces are also investigated (Theorems 3, 5 and 11). It is also shown that if A is a left brace whose cardinality is an odd number and (-a) b=-(ab) for all a, b A, then A is a two-sided brace and hence a Jacobson radical ring. It is also observed that the semidirect product and the wreath product of braces of a finite multipermutation level is a brace of a fi…
Extending structures of Rota-Baxter Lie algebras
本文通过向量空间引入Rota-Baxter Lie代数的扩展数据概念，构造统一积，并解决其扩展结构问题，从而推广并统一了已有的扩展问题与分解问题。
Quasi-triangular, triangular, factorizable anti-Leibniz bialgebras and anti-Leibniz Yang-Baxter equation
引入反Leibniz双代数及其与Manin triple和matched pair的等价性，提出反Leibniz Yang-Baxter方程并通过relative Rota-Baxter算子构造对称解，同时研究可分解结构与维数扩展。
Extending Structures for Rota-Baxter family Hom-associative Algebras
该研究针对 Rota-Baxter 族 Hom-结合代数，定义扩张数据、统一积、匹配对与形变映射，依次解决了扩张结构、因子分解及分类补三类理论问题，并给出旗数据相关应用实例。
This paper studies operator forms of the nonhomogeneous associative classical Yang-Baxter equation (nhacYBe), extending and generalizing such studies for the classical Yang-Baxter equation and associative Yang-Baxter equation that can be tracked back to the works of Semonov-Tian-Shansky and Kupershmidt on Rota-Baxter Lie algebras and $\mathcal{O}$-operators. In general, solutions of the nhacYBe are characterized in terms of generalized $\mathcal{O}$-operators. The characterization can be given by the classical $\mathcal{O}$-operators precisely when the solutions satisfy an invariant condition.…
Twisted Rota-Baxter family operators on Hom-associative algebras
本文定义 Hom-结合代数上的扭曲 Rota-Baxter 族算子，研究对应 Hom-NS-族代数、上同调与形变理论，并给出刚性的充分条件。
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