in particular will be helpful. Both results can be summarized in the statement that a connected cocommutative Hopf algebra is the enveloping algebra of a Lie algebra, as an algebra, and cofree, as a coalgebra. At any rate, we discuss a strong version of the Poincaré–Birkhoff–Witt theorem in Section 4, and the Cartier–Milnor–Moore theorem at the end of Section 5. The necessary notions of Rota–Baxter operator theory will be introduced and explained in due course.

Whereas the Hopf algebraic description springs up naturally from the intrinsic geometrical approach, we have found it expedient to smooth this transition with the help of Lie–Rinehart algebras: these constitute the “noncommutative geometry” version of Lie algebroid technology.

The plan of the work is as follows. In this section we explain our main aims and fix some notations of frequent use. Next we recommend a look at Appendix A, indispensable for everything that follows; most readers will just need to scan it for the notations. In Section 2 we address for the first time Lie–Scheffers systems; they are intimately linked to equations  and .

After this, two paths are possible: either reading Appendices C and D for motivation, or not. Most of the stuff in 