ALGEBRAIC STUDIES OF SYMMETRIC OPERATORS

Differential operators and integral operators are linked together by the first fundamental theorem of calculus. Based on this principle, the notion of a differential Rota-Baxter algebra was proposed by Guo and Keigher. Recently, the subject has attracted more attention since it is associated with many areas in mathematics, such as integro-differential algebras. This paper considers differential Rota-Baxter algebras in the quasi-idempotent operator context. We establish a Gröbner-Shirshov basis for free commutative quasi-idempotent differential algebras (resp. Rota-Baxter algebras, resp. differential Rota-Baxter algebras). This provides a linear basis of a free object in each of the three corresponding categories by the Composition-Diamond lemma. Communicated by P. Kolesnikov

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.

Construction of free differential algebras by extending Gröbner-Shirshov bases

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.

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.

In the paper, we nd the connection between solutions of the classical YangBaxter equation with an ad-invariant symmetric part and RotaBaxter operators of special type on a real general linear algebra gl n (R).Using this connection, we classify solutions of the classical YangBaxter equation with an ad-invariant symmetric part on gl 2 (C) using the classication of RotaBaxter operators of nonzero weight on gl 2 (C) and a classication of RotaBaxter operators of weight 0 on sl 2 (C).

This article discusses how associations form professional identities and reflects on the establishment of the Journal of Western Archives. It highlights the ways in which the establishment of the journal is a manifestation of the development of a Western regional archival identity and underscores the role that individuals play in the creation of associational identities. One of the most compelling vignettes in world literature, found in Miguel de Cervantes’ classic novel Don Quijote, deals with the difficulties of establishing identity. In the novel, Don Quijote forges a new identity for himself as a valiant knight errant; he is accompanied by his neighbor Sancho Panza, whom he has made his faithful squire, as he goes on various adventures. In the incident referenced, Quijote and Panza have just come upon thirty or forty windmills. Quijote declares that the windmills are giants and that they must be vanquished before they can cause more misery and suffering. The following conversation, illustrative of the difficulty of forming a new identity, occurs. “What giants?” asked Sancho Panza. “The ones you can see over there,” answered his master, “with the huge arms, some of which are very nearly two leagues long.” “Now look, your grace,” said Sancho, “what you see over there aren’t giants, but windmills, and what seem to be arms are just their sails, that go around in the wind and turn the millstone.” “Obviously,” replied Don Quijote, “you don’t know much about adventures. Those are giants—and if you’re frightened, 1 Daines: Tilting at Windmills Published by DigitalCommons@USU, 2010 take yourself away from here and say your prayers, while I go charging into savage and unequal combat with them.” Quijote, in reality a retired country gentleman named Alonzo Quijano, is clearly deeply invested in the new identity that he is creating for himself, while Sancho Panza is struggling to adapt to his new identity as a faithful squire. Panza is far more comfortable with his former identity as Quijano’s neighbor. Over the course of the novel, Panza comes to accept the new identity that Quijano has created for him and recognize the benefits that this new identity has given him. The process of identity formation is particularly interesting to me, as I have been deeply enmeshed, over the last two decades, in the process of transitioning from an identity of a student studying to enter medical school to the identity of a professional archivist. Identity is composed of various meanings that we and others attach to ourselves. These “meanings, or self-conceptions, are based on the social roles and group memberships a person holds (social identities) as well as the personal and character traits they display, and others attribute to them, based on their conduct (personal identities).” In professional settings, identity development “often involves tailoring one’s identity to a prototypic role identity.” This is a difficult task for archivists working in the United States, as our archival identity is currently in flux. Rand Jimerson has argued that the “most striking feature of the American archival profession in recent years is its ongoing search for identity and for public acceptance as a socially significant profession.” The archival profession’s concern with identity development is not unique. Other professions have also struggled to create cohesive professional identities that are seen as socially significant by the general public. Jimerson identifies three broad manifestations of American archivists’ search for professional identity as “developing internal standards for professional recognition; enhancing the public image of archives and archivists; and strengthening the research and theoretical foundations of the profession.” 1. Miguel de Cervantes Saavedra, Don Quijote: A New Translation, Backgrounds and Contexts, Criticism/ Miguel de Cervantes, trans. Burton Raffel; ed. Diana de Armas Wilson (New York: W. W. Norton, 1999), 43–44. 2. Herminia Ibarra and Jennifer L. Petriglieri, “Identity Work and Play,” Journal of Organizational Change Management 23, no. 1 (2010), 11.

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.

In this paper, we introduce an algebra structure denoted by InvDer algebra in which we twist an algebra with an invertible derivation whose inverse is also a derivation. We define InvDer Lie algebras, InvDer associative algebras, InvDer Zinbiel algebras and InvDer dendriforme algebras. We also study the relations between these structures by using Rota–Baxter operators and endomorphism operators.

This chapter studies, in the setting of vector spaces over a field, the basics concerning multilinear functions, tensor products, spaces of linear functions, and algebras related to tensor products. Sections 1–5 concern special properties of bilinear forms, all vector spaces being assumed to be finite-dimensional. Section 1 associates a matrix to each bilinear form in the presence of an ordered basis, and the section shows the effect on the matrix of changing the ordered basis. It then addresses the extent to which the notion of “orthogonal complement” in the theory of inner-product spaces applies to nondegenerate bilinear forms. Sections 2–3 treat symmetric and alternating bilinear forms, producing bases for which the matrix of such a form is particularly simple. Section 4 treats a related subject, Hermitian forms when the field is the complex numbers. Section 5 discusses the groups that leave some particular bilinear and Hermitian forms invariant. Section 6 introduces the tensor product of two vector spaces, working with it in a way that does not depend on a choice of basis. The tensor product has a universal mapping property—that bilinear functions on the product of the two vector spaces extend uniquely to linear functions on the tensor product. The tensor product turns out to be a vector space whose dual is the vector space of all bilinear forms. One particular application is that tensor products provide a basis-independent way of extending scalars for a vector space from a field to a larger field. The section includes a number of results about the vector space of linear mappings from one vector space to another that go hand in hand with results about tensor products. These have convenient formulations in the language of category theory as “natural isomorphisms.” Section 7 begins with the tensor product of three and then $n$ vector spaces, carefully considering the universal mapping property and the question of associativity. The section defines an algebra over a field as a vector space with a bilinear multiplication, not necessarily associative. If $E$ is a vector space, the tensor algebra $T(E)$ of $E$ is the direct sum over $n\geq0$ of the $n$-fold tensor product of $E$ with itself. This is an associative algebra with a universal mapping property relative to any linear mapping of $E$ into an associative algebra $A$ with identity: the linear map extends to an algebra homomorphism of $T(E)$ into $A$ carrying 1 into 1. Sections 8–9 define the symmetric and exterior algebras of a vector space $E$. The symmetric algebra $S(E)$ is a quotient of $T(E)$ with the following universal mapping property: any linear mapping of $E$ into a commutative associative algebra $A$ with identity extends to an algebra homomorphism of $S(E)$ into $A$ carrying 1 into 1. The symmetric algebra is commutative. Similarly the exterior algebra $\bigwedge(E)$ is a quotient of $T(E)$ with this universal mapping property: any linear mapping $l$ of $E$ into an associative algebra $A$ with identity such that $l(v)^2=0$ for all $v\in E$ extends to an algebra homomorphism of $\bigwedge(E)$ into $A$ carrying 1 into 1. The problems at the end of the chapter introduce some other algebras that are of importance in applications, and the problems relate some of these algebras to tensor, symmetric, and exterior algebras. Among the objects studied are Lie algebras, universal enveloping algebras, Clifford algebras, Weyl algebras, Jordan algebras, and the division algebra of octonions.

ABSTRACTIn this paper, we define pre-Malcev algebras and alternative quadri-algebras and prove that they generalize pre-Lie algebras and quadri-algebras, respectively, to the alternative setting. We use the results and techniques from [4, 14] to discuss and give explicit computations of different constructions in terms of bimodules, splitting of operations, and Rota–Baxter operators.

Rota–Baxter operators on Witt and Virasoro algebras

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.

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.

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.