Quantum Lie algebras of type A_n positive, PBW bases and the Yang-Baxter equation

We show explicitly a generalised Lie algebra embedded in the positive and negative parts of the Drinfeld-Jimbo quantum groups of type A_n. Such a generalised Lie algebra satisfy axioms closely related to the ones found by S.L. Woronowicz. For the universal enveloping algebra of such generalised Lie algebras we establish several conditions in order to obtain bases of type Poincar\'e-Birkhoff-Witt. Besides a graded algebra is proposed and some relations with the quantum Yang-Baxter equation are studied.