Rota-Baxter Research Articles

Free Lie Rota–Baxter algebras

Free Lie Rota–Baxter algebras

Rota–Baxter operators on Witt and Virasoro algebras

The homogeneous Rota-Baxter operators on the Witt and Virasoro algebras are classified. As applications, the induced solutions of the classical Yang-Baxter equation and the induced pre-Lie and PostLie algebra structures are obtained.

Rota–Baxter systems, dendriform algebras and covariant bialgebras

A generalisation of the notion of a Rota–Baxter operator is proposed. This generalisation consists of two operators acting on an associative algebra and satisfying equations similar to the Rota–Baxter equation. Rota–Baxter operators of any weights and twisted Rota–Baxter operators are solutions of the proposed system. It is shown that dendriform algebra structures of a particular kind are equivalent to Rota–Baxter systems. It is shown further that a Rota–Baxter system induces a weak pseudotwistor [Panaite and Van Oystaeyen (2015) [15]] which can be held responsible for the existence of a new associative product on the underlying algebra. Examples of solutions of Rota–Baxter systems are obtained from quasitriangular covariant bialgebras hereby introduced as a natural extension of infinitesimal bialgebras [Aguiar (2000) [3]].

Classification of monomial Rota–Baxter operators on k[x

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].

On Bradley's q-MZVs and a generalized Euler decomposition formula

We propose an algebra extension for Bradley's q-analogue of multiple zeta values and provide a double q-shuffle relation using Rota–Baxter operators for the q-shuffle product. In this setting we prove a generalized Euler decomposition formula.

Weighted Tensor Products of Joyal Species, Graphs, and Charades

Motivated by the weighted Hurwitz product on sequences in an algebra, we produce a family of monoidal structures on the category of Joyal species. We suggest a family of tensor products for charades. We begin by seeing weighted derivational algebras and weighted Rota-Baxter algebras as special monoids and special semigroups, respectively, for the same monoidal structure on the category of graphs in a monoidal additive category. Weighted derivations are lifted to the categorical level.

Yang–Baxter operators need quantum entanglement to distinguish knots

Any solution to the Yang–Baxter equation yields a family of representations of braid groups. Under certain conditions, identified by (Turaev 1988 Inventiones Math. 92 527–53), the appropriately normalized trace of these representations yields a link invariant. Any Yang–Baxter solution can be interpreted as a two-qudit quantum gate. Here we show that if this gate is non-entangling, then the resulting invariant of knots is trivial. We thus obtain a connection between topological entanglement and quantum entanglement.

Rota-Baxter algebras, singular hypersurfaces, and renormalization on Kausz compactifications

We consider Rota-Baxter algebras of meromorphic forms with poles along a (singular) hypersurface in a smooth projective variety and the associated Birkhoff factorization for algebra homomorphisms from a commutative Hopf algebra. In the case of a normal crossings divisor, the Rota-Baxter structure simplifies considerably and the factorization becomes a simple pole subtraction. We apply this formalism to the unrenormalized momentum space Feynman amplitudes, viewed as (divergent) integrals in the complement of the determinant hypersurface. We lift the integral to the Kausz compactification of the general linear group, whose boundary divisor is normal crossings. We show that the Kausz compactification is a Tate motive and that the boundary divisor and the divisor that contains the boundary of the chain of integration are mixed Tate configurations. The regularization of the integrals that we obtain differs from the usual renormalization of physical Feynman amplitudes, and in particular it may give mixed Tate periods in some cases that have non-mixed Tate contributions when computed with other renormalization methods.

Monads and distributive laws for Rota–Baxter and differential algebras

In recent years, algebraic studies of the differential calculus and integral calculus in the forms of differential algebra and Rota–Baxter algebra have been merged together to reflect the close relationship between the two calculi through the First Fundamental Theorem of Calculus. In this paper we study this relationship from a categorical point of view in the context of distributive laws which can be tracked back to the distributive law of multiplication over addition. The monad giving Rota–Baxter algebras and the comonad giving differential algebras are constructed. Then we obtain monads and comonads giving the composite structures of differential and Rota–Baxter algebras. As a consequence, a mixed distributive law of the monad giving Rota–Baxter algebras over the comonad giving differential algebras is established.

Rota–Baxter multiplicative 3-ary Hom-Nambu–Lie algebras

In this paper, we introduce the concepts of Rota–Baxter operators and differential operators with weights on a multiplicative n -ary Hom-algebra. We then focus on Rota–Baxter multiplicative 3-ary Hom-Nambu–Lie algebras and show that they can be derived from Rota–Baxter Hom-Lie algebras, Hom-preLie algebras and Rota–Baxter commutative Hom-associative algebras. We also explore the connections between these Rota–Baxter multiplicative 3-ary Hom-Nambu–Lie algebras.

Rota-Baxter type operators, rewriting systems and Gröbner-Shirshov bases

In this paper we apply the methods of rewriting systems and Grobner-Shirshov bases to give a unified approach to a class of linear operators on associative algebras. These operators resemble the classic Rota-Baxter operator, and they are called Rota-Baxter type operators. We characterize a Rota-Baxter type operator by the convergency of a rewriting system associated to the operator. By associating such an operator to a Grobner-Shirshov basis, we obtain a canonical basis for the free algebras in the category of associative algebras with that operator. This construction include as special cases several previous ones for free objects in similar categories, such as those of Rota-Baxter algebras and Nijenhuis algebras.

Rota–Baxter operators on the polynomial algebra, integration, and averaging operators

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.

Drinfeld basis for string-inspired Baxter operators

We propose Drinfeld's second realisation of the quantum group relevant to the Lax-operator approach developed in the work of Bazhanov, Frassek, Lukowski, Meneghelli and Staudacher.

A q-difference Baxter operator for the Ablowitz–Ladik chain

We construct the Baxter operator and the corresponding Baxter equation for a quantum version of the Ablowitz–Ladik model. The result is achieved in two different ways: by using the well-known Bethe ansatz technique and by looking at the quantum analogue of the classical Bäcklund transformations. General results about integrable models governed by the same r-matrix algebra will be given. Baxter’s equation comes out to be a q-difference equation involving both the trace and the quantum determinant of the monodromy matrix. The spectrality property of the classical Bäcklund transformations gives a trace formula representing the classical analogue of Baxter’s equation. A q-integral representation of the Baxter operator is discussed.

Cofree Hopf algebras on Hopf bimodule algebras

We investigate a Hopf algebra structure on the cotensor coalgebra associated to a Hopf bimodule algebra which contains universal version of Clifford algebras and quantum groups as examples. It is shown to be the bosonization of the quantum quasi-shuffle algebra built on the space of its right coinvariants. The universal property and a Rota–Baxter algebra structure are established on this new algebra.

Bimodules and Rota-Baxter Relations

In this paper we introduce and study Hom-type bimodules of some Hom-algebraic structures endowed with Rota-Baxter relations. We introduce bimodules over Homassociative Rota-Baxter algebras and give their various twisting and their connection with bimodules over Hom-preLie algebras. Then we introduce Rota-Baxter q- Homtridendriform algebras. Next we express axioms defining q-Hom-tridendriform algebras by mean of vector basis. Moreover we introduce bimodules over q-Homtridendriform algebras and give some examples, and prove that they are closed by twisting. Finally we give their connection with Hom-associative Rota-Baxter bimodules.

Generating functions from the viewpoint of Rota–Baxter algebras

We study generating functions in the context of Rota–Baxter algebras. We show that exponential generating functions can be naturally viewed as a very special case of complete free commutative Rota–Baxter algebras. This viewpoint allows us to apply free Rota–Baxter algebras to give a large class of algebraic structures in which generalizations of generating functions can be studied. We generalize the product formula and composition formula for exponential power series. We also give generating functions both for known number families such as Stirling numbers of the second kind and partition numbers, and for new number families such as those from not necessarily disjoint partitions and partitions of multisets.

Free integro-differential algebras and Gröbner–Shirshov bases

The notion of commutative integro-differential algebra was introduced for the algebraic study of boundary problems for linear ordinary differential equations. Its noncommutative analog achieves a similar purpose for linear systems of such equations. In both cases, free objects are crucial for analyzing the underlying algebraic structures, e.g. of the (matrix) functions.In this paper we apply the method of Gröbner–Shirshov bases to construct the free (noncommutative) integro-differential algebra on a set. The construction is from the free Rota–Baxter algebra on the free differential algebra on the set modulo the differential Rota–Baxter ideal generated by the noncommutative integration by parts formula. In order to obtain a canonical basis for this quotient, we first reduce to the case when the set is finite. Then in order to obtain the monomial order needed for the Composition-Diamond lemma, we consider the free Rota–Baxter algebra on the truncated free differential algebra. A Composition-Diamond lemma is proved in this context, and a Gröbner–Shirshov basis is found for the corresponding differential Rota–Baxter ideal.

The Pre-Lie Structure of the Time-Ordered Exponential

The usual time-ordering operation and the corresponding time-ordered exponential play a fundamental role in physics and applied mathematics. In this work we study a new approach to the understanding of time-ordering relying on recent progress made in the context of enveloping algebras of pre-Lie algebras. Various general formulas for pre-Lie and Rota-Baxter algebras are obtained in the process. Among others, we recover the noncommutative analog of the classical Bohnenblust-Spitzer formula, and get explicit formulae for operator products of time-ordered exponentials.

Construction of Rota-Baxter algebras via Hopf module algebras

We propose the notion of Hopf module algebras and show that the projection onto the subspace of coinvariants is an idempotent Rota-Baxter operator of weight -1. We also provide a construction of Hopf module algebras by using Yetter-Drinfeld module algebras. As an application, we prove that the positive part of a quantum group admits idempotent Rota-Baxter algebra structures.