Groebner-Shirshov basis of the universal enveloping Rota-Baxter algebra of a Lie algebra

Abstract

Consider the class RBLie of Lie algebras equipped with a Rota---Baxter operator. Then the forgetful functor RBLie --> Lie has a left adjoint one denoted by $U_{RB}(\cdot)$. We prove an "operator" analogue of the Poincare---Birkhoff---Witt theorem for $U_{RB}(L)$, where $L$ is an arbitrary Lie algebra, by means of Gr\"obner---Shirshov bases theory for Lie algebras with an additional operator.