Rota–Baxter Operators on Quadratic Algebras

Abstract

We prove that all Rota-Baxter operators on a quadratic division algebra are trivial. For nonzero weight, we state that all Rota-Baxter operators on the simple odd-dimensional Jordan algebra of bilinear form are projections on a subalgebra along another one. For weight zero, we find a connection between the Rota-Baxter operators and the solutions to the alternative Yang-Baxter equation on the Cayley-Dickson algebra. We also investigate the Rota-Baxter operators on the matrix algebras of order two, the Grassmann algebra of plane, and the Kaplansky superalgebra.Mathematics Subject Classification. 16T25, 17A45, 17C50.

Rota–Baxter operators of nonzero weight on the matrix algebra of order three

“…. + k v and for A = (a cd ) n c,d=1 = R(x), we get the equalities a ij (λ u − λ v ) = a ij (λ p − λ q ) by (7). When u − v = t, we get a ij = 0.…”

“…Statement 4 [7]. Let A be a unital algebra, and let P be an RB-operator of nonzero weight λ on A. a) If P (1) ∈ F , then P is splitting; b) If P (P (x) + λx) = 0 for all x ∈ A, then P is splitting.…”

“…Now we want to use Lemma 1 to specify the information about R. Let us illustrate the statement of Lemma 1 in this case in details. We will need the equation (7). Let a ∈ M 3 (F ) and let R(a) = (α ij ).…”

“…In §3, we consider some results about RB-operators of nonzero weight on the matrix algebra. In Corollary 1, we clarify the classification of RB-operators on M 2 (C) from [7] .…”

Rota–Baxter operators on a sum of fields

“…Let us start with some basic properties of Rota-Baxter operators. Lemma 1 [6,4]. Let A be an algebra and let P be an RB-operator of weight λ on A.…”