Splitting the Goldman-Turaev Lie bialgebra along a simple separating curve

We introduce the notion of a double Lie bimodule, consisting of a Lie bialgebra L, a Lie L-bimodule P, and a double Lie bracket on P satisfying suitable compatibility conditions. We describe an algebraic construction that combines two double Lie bimodules into a Lie bialgebra. As a geometric application, we show that the Goldman-Turaev Lie bialgebra of a surface with boundary, together with the Kawazumi-Kuno intersection operations on the linear span of the set of homotopy classes of paths between pairwise distinct boundary points, may be assembled into a double Lie bimodule. We prove a decomposition theorem for the Goldman-Turaev Lie bialgebra of an oriented surface in terms of the double Lie bimodules corresponding to the two surfaces with boundary obtained by cutting along a simple separating closed curve.