Cocycle Hopf Algebra Structures on Free Modified Rota–Baxter Algebras by Vertex-decorated Rooted Trees

Abstract

A modified Rota–Baxter operator originates from the convolution theorem for the Hilbert transformation by Tricomi. It also satisfies the modified classical Yang–Baxter equation discovered by Semenov-Tian-Shansky, and is applied to Lax equations and affine geometry of Lie groups later. In this paper, we provide an alternative explicit construction of free modified Rota–Baxter associative algebras from the perspective of combinatorial objects. First, we revisit a construction of free operated bialgebra structure on vertex-decorated rooted forests via a variant of the Hochschild 1-cocycle condition. Applying an isomorphism between two kinds of free operated algebras given by different carriers, we construct free modified Rota–Baxter algebras on vertex-decorated rooted forests. We then obtain a combinatorial coproduct on the free modified Rota–Baxter algebra by means of the universal property of free operated algebras, leading to a cocycle bialgebra structure and further a cocycle Hopf algebra structure on it.