𝒪-operators on associative algebras and associative Yang–Baxter equations

Chengming Bai, Li Guo and Xiang Ni

Vol. 256 (2012), No. 2, 257–289

DOI: 10.2140/pjm.2012.256.257

Abstract

An 𝒪-operator on an associative algebra is a generalization of a Rota–Baxter operator that plays an important role in the Hopf algebra approach of Connes and Kreimer to the renormalization of quantum field theory. It is also the associative analog of an 𝒪-operator on a Lie algebra in the study of the classical Yang–Baxter equation. We introduce the concept of an extended 𝒪-operator on an associative algebra whose Lie algebra analog has been applied to generalized Lax pairs and PostLie algebras. We study algebraic structures coming from extended 𝒪-operators. Continuing the work of Aguiar deriving Rota–Baxter operators from the associative Yang–Baxter equation, we show that its solutions correspond to extended 𝒪-operators through a duality. We also establish a relationship of extended 𝒪-operators with the generalized associative Yang–Baxter equation.

Keywords

𝒪-operator, Rota–Baxter operator, Yang–Baxter equation, bimodule

Mathematical Subject Classification 2010

Primary: 16W99, 17A30

Secondary: 57R56

Milestones

Received: 14 April 2011

Revised: 5 August 2011

Accepted: 27 September 2011

Published: 30 May 2012