Rota–Baxter operators on 4-dimensional complex simple associative algebras

Abstract

Rota–Baxter operators or relations were introduced to solve certain analytic and combinatorial problems and then applied to many fields in mathematics and mathematical physics. In this paper, we commence to study the Rota–Baxter operators of weight zero on a 4-dimensional complex simple associative algebra, which is isomorphic to the algebra of all 2×2 matrices over the field of complex numbers. Such operators satisfy (the operator form of) the classical Yang–Baxter equation on the general linear Lie algebra gl2(C). We provide two different solving methods: one by hand to show some tricks for solving a large nonlinear system, and another one by Computer Algebra to show the possibility of solving higher dimensional cases.