Rota–Baxter Operators on Unital Algebras

Rota–Baxter operators on a sum of fields

“…We continue investigations fulfilled by S. de Bragança in 1975 [6] and by H. An and C. Bai in 2008 [1]. Since all RB-operators on A of weight zero are trivial [12], i.e., equal to 0, we study only RB-operators on A of nonzero weight λ.…”

“…Proof. If λ = 0, then R = 0 [12] and x • R y = 0. For λ = 0, we may assume that λ = 1, since rescalling of the product does not exchange the algebraic structure.…”

“…G. Baxter introduced the notion of a Rota-Baxter operator in 1960 [3] as a natural generalization of integration by parts formula for the integral operator. Further, many authors including G.-C. Rota, P. Cartier, L. Guo have studied RB-operators, see details in [6,5]. There are deep connections of Rota-Baxter algebras with mathematical physics, number theory, operad theory, combinatorics, and others.…”

“…In the study of RB-operators, the presence of the multiplicative identity (unit) in the algebra is a crucial condition. As shown in [5], RB-operators on a unital algebra subject to various additional restrictions in comparison with non-unital algebras.…”

An example of a simple double Lie algebra

“…This notion firstly appeared in the article [21] of F. Tricomi in 1951 and further was several times rediscovered [5,6], see the monograph [10]. Let us mention the bijection [1,9,18] between RB-operators of weight 0 on the matrix algebra and its solutions of the associative Yang Baxter equation (AYBE) [3,17,24]. We state the correspondence between countable-dimensional double Lie algebras V and Rota Baxter operators acting from the space of finitary matrices [4] to End(V ) and satisfying some additional finiteness conditions.…”