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

Abstract

Abstract. 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 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.

Simple finite-dimensional double algebras

“…Skew-symmetric Rota-Baxter operators are in one-to-one correspondence with constant solutions of the classical Yang-Baxter equation, see [9]. The second relation in (11) is known as averaging equation. Algebras with such operators (averaging algebras) are of substantial interest in functional analysis, they have also been studied from combinatorial point of view [8,11].…”

“…The second relation in (11) is known as averaging equation. Algebras with such operators (averaging algebras) are of substantial interest in functional analysis, they have also been studied from combinatorial point of view [8,11].…”

Nonabelian embedding tensors

“…More generally, averaging operators are studied on the algebras over the general operad P and give rise to the di-P algebras and tri-P algebras [12]. Averaging operators are related to the procedure of replication, Hadamard products and Manin white products in the operad theory [12,24,25,31], and can be applied to study double algebras, classical Yang-Baxter equations and Gröbner-Shirshov bases [10,11,13,33]. Note that the above vector space V in the definition of an embedding tensor is understood as the space that 1-form fields take values in.…”

Rota-type operators on 3-dimensional nilpotent associative algebras

“…(1) some other algebras [4], [11], [16]. The study of some particular cases of Rota-Baxter operators was initiated in [22], [23] and [8]. Recently, in [13] the authors presented all homogeneous Rota-type operators on null-filiform associative algebras.…”