Rota-Baxter Research Articles

Time-Ordering and a Generalized Magnus Expansion

Both the classical time-ordering and the Magnus expansion are well-known in the context of linear initial value problems. Motivated by the noncommutativity between time-ordering and time derivation, and related problems raised recently in statistical physics, we introduce a generalization of the Magnus expansion. Whereas the classical expansion computes the logarithm of the evolution operator of a linear differential equation, our generalization addresses the same problem, including however directly a non-trivial initial condition. As a by-product we recover a variant of the time ordering operation, known as T*-ordering. Eventually, placing our results in the general context of Rota-Baxter algebras permits us to present them in a more natural algebraic setting. It encompasses, for example, the case where one considers linear difference equations instead of linear differential equations.

Embedding of dendriform algebras into Rota-Baxter algebras

Abstract Following a recent work [Bai C., Bellier O., Guo L., Ni X., Splitting of operations, Manin products, and Rota-Baxter operators, Int. Math. Res. Not. IMRN (in press), DOI: 10.1093/imrn/rnr266] we define what is a dendriform dior trialgebra corresponding to an arbitrary variety Var of binary algebras (associative, commutative, Poisson, etc.). We call such algebras di- or tri-Var-dendriform algebras, respectively. We prove in general that the operad governing the variety of di- or tri-Var-dendriform algebras is Koszul dual to the operad governing di- or trialgebras corresponding to Var!. We also prove that every di-Var-dendriform algebra can be embedded into a Rota-Baxter algebra of weight zero in the variety Var, and every tri-Var-dendriform algebra can be embedded into a Rota-Baxter algebra of nonzero weight in Var.

Nijenhuis algebras, NS algebras, and N-dendriform algebras

In this paper, we study (associative) Nijenhuis algebras, with emphasis on the relationship between the category of Nijenhuis algebras and the categories of NS algebras and related algebras. This is in analogy to the well-known theory of the adjoint functor from the category of Lie algebras to that of associative algebras, and the more recent results on the adjoint functor from the categories of dendriform and tridendriform algebras to that of Rota-Baxter algebras. We first give an explicit construction of free Nijenhuis algebras and then apply it to obtain the universal enveloping Nijenhuis algebra of an NS algebra. We further apply the construction to determine the binary quadratic nonsymmetric algebra, called the N-dendriform algebra, that is compatible with the Nijenhuis algebra. As it turns out, the N-dendriform algebra has more relations than the NS algebra.

Differential type operators and Gröbner–Shirshov bases

A long standing problem of Gian-Carlo Rota for associative algebras is the classification of all linear operators that can be defined on them. In the 1970s, there were only a few known operators, for example, the derivative operator, the difference operator, the average operator, and the Rota–Baxter operator. A few more appeared after Rota posed his problem. However, little progress was made to solve this problem in general. In part, this is because the precise meaning of the problem is not so well understood. In this paper, we propose a formulation of the problem using the framework of operated algebras and viewing an associative algebra with a linear operator as one that satisfies a certain operated polynomial identity. This framework also allows us to apply theories of rewriting systems and Gröbner–Shirshov bases. To narrow our focus more on the operators that Rota was interested in, we further consider two particular classes of operators, namely, those that generalize differential or Rota–Baxter operators. These two classes of operators correspond to those that possess Gröbner–Shirshov bases under two different monomial orderings. Using this framework and computer algebra, we are able to come up with a list of these two classes of operators, and provide some evidence that these lists may be complete. Our search has revealed quite a few new operators of these types whose properties are expected to be similar to the differential operator and Rota–Baxter operator respectively.Recently, a more unified approach has emerged in related areas, such as difference algebra and differential algebra, and Rota–Baxter algebra and Nijenhuis algebra. The similarities in these theories can be more efficiently explored by advances on Rotaʼs problem.

Splitting of Operations, Manin Products, and Rota–Baxter Operators

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.

PostLie algebra structures on the Lie Algebra SL(2,Ca)

The PostLie algebra is an enriched structure of the Lie algebra that has recently arisen from operadic study. It is closely related to pre-Lie algebra, Rota-Baxter algebra, dendriform trialgebra, modified classical Yang-Baxter equations and integrable systems. This paper gives a complete classification of PostLie algebra structures on the Lie algebra sl(2,C) up to isomorphism. The classification problem is first reduced to solving an equation of 3 × 3 matrices. Then the latter problem is solved by making use of the classification of complex symmetric matrices up to the congruent action of orthogonal groups.

Baxter operators for arbitrary spin

We construct Baxter operators for the homogeneous closed XXX spin chain with the quantum space carrying infinite- or finite-dimensional s ℓ 2 representations. All algebraic relations of Baxter operators and transfer matrices are deduced uniformly from Yang–Baxter relations of the local building blocks of these operators. This results in a systematic and very transparent approach where the cases of finite- and infinite-dimensional representations are treated in analogy. Simple relations between the Baxter operators of both cases are obtained. We represent the quantum spaces by polynomials and build the operators from elementary differentiation and multiplication operators. We present compact explicit formulae for the action of Baxter operators on polynomials.

Baxter operators for arbitrary spin II

This paper presents the second part of our study devoted to the construction of Baxter operators for the homogeneous closed XXX spin chain with the quantum space carrying infinite or finite-dimensional s ℓ 2 representations. We consider the Baxter operators used in Bazhanov et al. (1996, 1997, 1999, 2010) [1,2], formulate their construction uniformly with the construction of our previous paper. The building blocks of all global chain operators are derived from the general Yang–Baxter operators and all operator relations are derived from general Yang–Baxter relations. This leads naturally to the comparison of both constructions and allows to connect closely the treatment of the cases of infinite-dimensional representation of generic spin and finite-dimensional representations of integer or half-integer spin. We prove not only the relations between the operators but present also their explicit forms and expressions for their action on polynomials representing the quantum states.

Embedding dendriform algebra into its universal enveloping Rota-Baxter algebra

In this paper, by using Gröbner-Shirshov bases for Rota-Baxter algebras, we prove that every dendriform algebra over a field of characteristic 0 can be embedded into its universal enveloping Rota-Baxter algebra.

𝒪-Operators of Loday Algebras and Analogues of the Classical Yang–Baxter Equation

We introduce notions of 𝒪-operators of the Loday algebras including the dendriform algebras and quadri-algebras as a natural generalization of Rota–Baxter operators. The invertible 𝒪-operators give a sufficient and necessary condition on the existence of the 2 n+1 operations on an algebra with the 2 n operations in an associative cluster. The analogues of the classical Yang–Baxter equation in these algebras can be understood as the 𝒪-operators associated to certain dual bimodules. As a byproduct, the constraint conditions (invariances) of nondegenerate bilinear forms on these algebras are given.

Gröbner-Shirshov bases for Rota-Baxter algebras

We establish the composition-diamond lemma for associative nonunitary Rota-Baxter algebras of weight λ. To give an application, we construct a linear basis for a free commutative and nonunitary Rota-Baxter algebra, show that every countably generated Rota-Baxter algebra of weight 0 can be embedded into a two-generated Rota—Baxter algebra, and prove the 1-PBW theorems for dendriform dialgebras and trialgebras.

Enumeration and Generating Functions of Differential Rota–Baxter Words

In this paper, the methods and results in enumeration and generation of Rota–Baxter words in Guo and Sit (Algebraic and Algorithmic Aspects of Differential and Integral Operators (AADIOS), Math. Comp. Sci., vol. 4, Sp. Issue (2,3), 2011) are generalized and applied to a free, non-commutative, non-unitary, ordinary differential Rota–Baxter algebra with one generator. A differential Rota–Baxter algebra is an associative algebra with two operators modeled after the differential and integral operators, which are related by the First Fundamental Theorem of Calculus. Differential Rota–Baxter words are words formed by concatenating differential monomials in the generator with images of words under the Rota–Baxter operator. Their totality is a canonical basis of a free, non-commutative, non-unitary, ordinary differential Rota–Baxter algebra. A free differential Rota–Baxter algebra can be constructed from a free Rota–Baxter algebra on a countably infinite set of generators. The order of the derivation gives another dimension of grading on differential Rota–Baxter words, enabling us to generalize and refine results from Guo and Sit to enumerate the set of differential Rota–Baxter words and outline an algorithm for their generation according to a multi-graded structure. Enumeration of a basis is often a first step to choosing a data representation in implementation of algorithms involving free algebras, and in particular, free differential Rota–Baxter algebras and several related algebraic structures on forests and trees. The generating functions obtained can be used to provide links to other combinatorial structures.

Enumeration and Generating Functions of Rota–Baxter Words

In this paper, we prove results on enumerations of sets of Rota–Baxter words ( $${{{\tt RBWs}}}$$ ) in a single generator and one unary operator. Examples of operators are integral operators, their generalization to Rota–Baxter operators, and Rota–Baxter type operators. $${{{\tt RBWs}}}$$ are certain words formed by concatenating generators and images of words under the operators. Under suitable conditions, they form canonical bases of free Rota–Baxter type algebras which are studied recently in relation to renormalization in quantum field theory, combinatorics, number theory, and operads. Enumeration of a basis is often a first step to choosing a data representation in implementation. We settle the case of one generator and one operator, starting with the idempotent case (more precisely, the exponent 1 case). Some integer sequences related to these sets of $${{{\tt RBWs}}}$$ are known and connected to other sequences from combinatorics, such as the Catalan numbers, and others are new. The recurrences satisfied by the generating series of these sequences prompt us to discover an efficient algorithm to enumerate the canonical basis of certain free Rota–Baxter algebras.

From Archimedean L-Factors to Topological Field Theories

We give a short account of our recent results on establishing a relation between topological field theories and Archimedean (∞-adic) geometry. First, we introduce a universal integral Baxter operator as a certain element of the spherical Hecke algebra \({\fancyscript{H}(G,K)}\). Eigenvalues of the Baxter operator acting on spherical vectors in irreducible representations G are given by corresponding local Archimedean L-factors. We provide a quantum field theory interpretation of the local Archimedean L-factors as certain correlation functions in a two-dimensional type A equivariant topological linear sigma model on a disk D. Next, we introduce a q-version of the Archimedean L-factors and construct a particular representation of q-deformed Whittaker functions generalizing the Casselman–Shalika formula for p-adic class one Whittaker functions. This representation allows to formulate a q-version of Archimedean Langlands correspondence. Then, we construct a representation of q-version of the Archimedean L-factors as a certain correlation functions in three-dimensional equivariant linear sigma model on D × S1. Finally, we propose an interpretation of the Archimedean Langlands correspondence as a mirror symmetry in the underlying topological quantum field theory.

Polylogarithms and multiple zeta values from free Rota-Baxter algebras

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.

COMPOSITION-DIAMOND LEMMA FOR λ-DIFFERENTIAL ASSOCIATIVE ALGEBRAS WITH MULTIPLE OPERATORS

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.

0-dialgebras with bar-unity and nonassociative Rota-Baxter algebras

We describe all homogeneous structures of Rota-Baxter algebras on a 0-dialgebra with associative bar-unity and give a corollary on the structure of a Rota-Baxter algebra on an arbitrary associative dialgebra with bar-unity as well as a unital associative conformal algebra. We prove that an arbitrary alternative dialgebra may be embedded into an alternative dialgebra with associative barunity. We suggest the definition of variety of dialgebras in the sense of Eilenberg which is equivalent to that introduced earlier by Kolesnikov.

SOME QUANTUM VERTEX ALGEBRAS OF ZAMOLODCHIKOV–FADDEEV TYPE

This is a continuation of a previous study of quantum vertex algebras of the Zamolodchikov–Faddeev type. In this paper, we focus our attention on the special case associated to diagonal unitary rational quantum Yang–Baxter operators. We prove that the associated weak quantum vertex algebras, if not zero, are irreducible quantum vertex algebras with a normal basis in a certain sense.

Generalized Shuffles Related to Nijenhuis and TD-Algebras

Shuffle type products are well known in mathematics and physics. They are intimately related to Loday's dendriform algebras and were extensively used to give explicit constructions of free 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, the so-called left-shift shuffle, can be used to give an explicit construction of the free unital commutative TD-algebra. We explore some basic properties of TD-operators. Our construction is related to Loday's unital commutative tridendriform algebra, including the involutive case. The concept of Rota–Baxter, Nijenhuis and TD-bialgebras is introduced at the end, and we show that any commutative bialgebra provides such objects.

ROTA–BAXTER OPERATORS ON GENERALIZED POWER SERIES RINGS

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.