Rota-Baxter Research Articles

Cohomology and Deformations of Relative Rota–Baxter Operators on Lie-Yamaguti Algebras

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.

Super-bimodules and $ \mathcal{O} $-operators of Bihom-Jordan superalgebras

<p>In this paper, we mainly study super-bimodules on Bihom-Jordan superalgebras and present some interesting constructions from the perspective of super-bimodules. Meanwhile, we give abelian extension through super-bimodules. In addition, we give the representations, $ \mathcal{O} $-operators and Rota–Baxter operators of Bihom-Jordan superalgebras. Specially, using $ \mathcal{O} $-operators, we characterize Bihom-pre-Jordan superalgebras.</p>

Hexa-algebras and related algebraic structures

In this paper, we introduce the notion of hexa-algebra. These are associative algebras where the product can be decomposed as the sum of six operations satisfying 21 axioms. We present the quasi-shuffle algebra and the tensor product of a dendriform and a tridendriform algebra as examples of hexa-algebras. We study the relation with ennea algebras and with other algebraic structures and we construct hexa-algebra structures from Rota–Baxter operators.

Matching Rota–Baxter modules and matching 𝒪-operators of associative algebras

In this paper, firstly, we introduce the notion of matching Rota–Baxter modules, which generalizes the notions of matching Rota–Baxter algebras and Rota–Baxter modules, and give its characteristic and construction. Secondly, we give the relation between matching Rota–Baxter modules and other modules structure. Finally, we introduce the notion of matching [Formula: see text]-operators of associative algebras which also generalizes the notion of matching Rota–Baxter algebras and give the solution of polarized associative Yang–Baxter equation by using the matching [Formula: see text]-operators of associative algebras. In addition, we introduce the definitions of the compatible matching [Formula: see text]-operators of associative algebras, matching Nijenhui operators, and compatible matching dendriform algebras, and consider their connection.

Post-Hopf algebras, relative Rota–Baxter operators and solutions to the Yang–Baxter equation

In this paper, first, we introduce the notion of post-Hopf algebra, which gives rise to a post-Lie algebra on the space of primitive elements and the fact that there is naturally a post-Hopf algebra structure on the universal enveloping algebra of a post-Lie algebra. A novel property is that a cocommutative post-Hopf algebra gives rise to a generalized Grossman–Larson product, which leads to a subadjacent Hopf algebra and can be used to construct solutions to the Yang–Baxter equation. Then, we introduce the notion of relative Rota–Baxter operator on Hopf algebras. A cocommutative post-Hopf algebra gives rise to a relative Rota–Baxter operator on its subadjacent Hopf algebra. Conversely, a relative Rota–Baxter operator also induces a post-Hopf algebra. Finally, we show that relative Rota–Baxter operators give rise to matched pairs of Hopf algebras. Consequently, post-Hopf algebras and relative Rota–Baxter operators give solutions to the Yang–Baxter equation in certain cocommutative Hopf algebras.

Hypergeometric identities related to Ruijsenaars systems

We present a proof of hypergeometric identities which play a crucial role in the theory of Baxter operators in the Ruijsenaars model.

Relative Rota--Baxter operators of arbitrary weight on Leibniz algebras and post-Leibniz algebra structures

Relative Rota--Baxter operators of arbitrary weight on Leibniz algebras and post-Leibniz algebra structures

RB-operator and Nijenhuis operator on Hom-associative conformal algebra

Hom-associative conformal algebra [Formula: see text] is an associative conformal algebra with a twist map and satisfies the Hom-associative conformal identity. This study aims to introduce the notion of the Rota–Baxter operator [Formula: see text] on Hom-associative conformal algebra [Formula: see text]. We generalize our study to Hom-dendriform and Hom-tridendriform conformal algebras and give their relation to Hom-preLie conformal algebras. We give the interrelation between dendriform (and tridendriform) algebra with Hom-associative conformal Rota–Baxter algebra. Furthermore, we explore the Nijenhus operator on Hom-associative conformal algebra and describe its relation with Hom-associative Rota–Baxter operator.

Monomial Rota–Baxter Operators of Nonzero Weight on F[x, y] Coming from Averaging Operators

Monomial Rota–Baxter Operators of Nonzero Weight on F[x, y] Coming from Averaging Operators

Free weighted (modified) differential algebras, free (modified) Rota–Baxter algebras and Gröbner–Shirshov bases

In this paper, we obtain, respectively, some new linear bases of free nonunitary (modified) weighted differential algebras and free nonunitary (modified) Rota–Baxter algebras, in terms of the method of Gröbner–Shirshov bases.

Derimorphisms over Algebras and Applications

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.

Representations of the Laurent series Rota-Baxter algebras and regular-singular decompositions

There is a Rota-Baxter algebra structure on the field A=k((t)) with P being the projection map from A=k[[t]]⊕t−1k[t−1] onto k[[t]]. We study the representation theory and regular-singular decompositions of any finite dimensional A-vector space. The main result shows that the category of finite dimensional representations is semisimple and consists of exactly three isomorphism classes of irreducible representations which are all one-dimensional. As a consequence, the number of GLA(V)-orbits in the set of all regular-singular decompositions of an n-dimensional A-vector space V is (n+1)(n+2)/2. We also use the result to compute the generalized class number, i.e., the number of the GLn(A)-isomorphism classes of finitely generated k[[t]]-submodules of An.

Rota–Baxter operators on groups

The theory of Rota–Baxter operators on rings and algebras has been developed since 1960. In 2020, the notion of Rota–Baxter operator on a group was defined. Further, it was proved that one may define a skew left brace on any group endowed with a Rota–Baxter operator. Thus, a group endowed with a Rota–Baxter operator gives rise to a set-theoretical solution to the Yang–Baxter equation. We provide some general constructions of Rota–Baxter operators on a group. Given a map on a group, we study its extensions to a Rota–Baxter operator. We state the connection between Rota–Baxter operators on a group and Rota–Baxter operators on an associated Lie ring. We describe Rota–Baxter operators on sporadic simple groups.

Rota–Baxter operators on Clifford semigroups and the Yang–Baxter equation

In this paper, we introduce the theory of Rota–Baxter operators on Clifford semigroups, useful tools for obtaining dual weak braces, i.e., triples (S,+,∘) where (S,+) and (S,∘) are Clifford semigroups such that a∘(b+c)=a∘b−a+a∘c and a∘a−=−a+a, for all a,b,c∈S. To each algebraic structure is associated a set-theoretic solution of the Yang–Baxter equation that has a behaviour near to the bijectivity and non-degeneracy. Drawing from the theory of Clifford semigroups, we provide methods for constructing dual weak braces and deepen some structural aspects, including the notion of ideal.

Free Rota-Baxter Family Algebras and Free (tri)dendriform Family Algebras

In this paper, we first construct the free Rota-Baxter family algebra generated by some set X in terms of typed angularly X-decorated planar rooted trees. As an application, we obtain a new construction of the free Rota-Baxter algebra only in terms of angularly decorated planar rooted trees (not forests), which is quite different from the known construction via angularly decorated planar rooted forests by K. Ebrahimi-Fard and L. Guo. We then embed the free dendriform (resp. tridendriform) family algebra into the free Rota-Baxter family algebra of weight zero (resp. one). Finally, we prove that the free Rota-Baxter family algebra is the universal enveloping algebra of the free (tri)dendriform family algebra.

Homotopy Rota–Baxter operators and post-Lie algebras

Rota–Baxter operators and the more general \mathcal{O} -operators, together with their interconnected pre-Lie and post-Lie algebras, are important algebraic structures, with Rota–Baxter operators and pre-Lie algebras instrumental in the Connes–Kreimer approach to renormalization of quantum field theory. This paper introduces the notions of a homotopy Rota–Baxter operator and a homotopy \mathcal{O} -operator on a symmetric graded Lie algebra. Their characterization by Maurer–Cartan elements of suitable differential graded Lie algebras is provided. Through the action of a homotopy \mathcal{O} -operator on a symmetric graded Lie algebra, we arrive at the notion of an operator homotopy post-Lie algebra, together with its characterization in terms of Maurer–Cartan elements. A cohomology theory of post-Lie algebras is established, with an application to 2-term skeletal operator homotopy post-Lie algebras.

λ-TD algebras, generalized shuffle products and left counital Hopf algebras

Operated algebras, that is, algebras equipped with linear operators, have important applications in mathematics and physics. Two primary instances of operated algebras are the Rota–Baxter algebra and TD-algebra. In this paper, we introduce a [Formula: see text]-TD algebra that includes both the Rota–Baxter algebra and the TD-algebra. The explicit construction of free commutative [Formula: see text]-TD algebra on a commutative algebra is obtained by a generalized shuffle product, called the [Formula: see text]-TD shuffle product. We then show that the free commutative [Formula: see text]-TD algebra possesses a left counital bialgebra structure by means of a suitable 1-cocycle condition. Furthermore, the classical result that every connected filtered bialgebra is a Hopf algebra, is extended to the context of left counital bialgebras. Given this result, we finally prove that the left counital bialgebra on the free commutative [Formula: see text]-TD algebra is connected and filtered, and thus is a left counital Hopf algebra.

Rota---Baxter operators on $Cur(sl_2(\mathbb{C}))$

We classify all Rota---Baxter operators on the simple Lie conformal algebra $\Cur(\sl_2(\mathbb{C}))$ and clarify which of them arise from the solutions to the conformal classical Yang---Baxter equation due to the connection discovered by Y. Hong and C. Bai in 2020.

Typed angularly decorated planar rooted trees and generalized Rota–Baxter algebras

We introduce a generalization of parametrized Rota–Baxter algebras, named \(\Omega \)-Rota–Baxter algebra, which includes family and matching Rota–Baxter algebras. We study the structure needed on the set \(\Omega \) of parameters in order to obtain that free \(\Omega \)-Rota–Baxter algebras are described in terms of typed and angularly decorated planar rooted trees: we obtain the notion of \(\lambda \)-extended diassociative semigroup, which includes sets (for matching Rota–Baxter algebras) and semigroups (for family Rota–Baxter algebras), and many other examples. We also describe free commutative \(\Omega \)-Rota–Baxter algebras generated by a commutative algebra A in terms of typed words.

Cohomology and deformations of weighted Rota–Baxter operators

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.