By Dixon M., Kurdachenko L., Subbotin I.

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Kh], see also [R2]. The set π(GI ) is a basis of H. Notice that (a) If u ∈ GI and u = vw, then v, w ∈ GI . (b) Any word u ∈ GI factorizes uniquely as a non-increasing product of Lyndon words in GI . On Nichols Algebras with Generic Braiding 53 Towards ﬁnding a PBW-basis the quotient H of T (V ), we look at the set SI := GI ∩ L. (6) We then deﬁne the function hI : SI → {2, 3, . . } ∪ {∞} by hI (u) := min t ∈ N : ut ∈ kX ut +I . (7) With these conventions, we are now able to state the main result of this subsection.

X Denote by τi : X → X the canonical projection for every i ∈ N. i Then the following sequence is exact. Xa ⊗ Xb ∇[(ξa ⊗ξb )a+b=n+1 ] −→ X ⊗X ∆[(τa ⊗τb )a+b=n ] −→ a+b=n+1 a+b=n X X ⊗ . Xa Xb Proof. 1, it remains to prove that the following sequence is exact Xa ⊗ Xb n ⊗ξbn )a+b=n+1 ] ∇[(ξa −→ Xn ⊗ Xn ∆[(τan ⊗τbn )a+b=n ] −→ a+b=n+1 a+b=n Xn Xa ⊗ Xn Xb Denote by γu : Xn Xn ⊗ → Xu Xn−u a+b=n Xn Xn ⊗ Xa Xb the canonical inclusion for every 0 ≤ u ≤ n. 1, in order to conclude we will prove that n γt ωtn ξun σuu−1 ⊗ ωn−t ξvn σvv−1 ∇ 0≤t≤n 1≤u,v≤n, u+v≥n+2 = ∇ ∆ is a monomorphism.

PBW-basis on the tensor algebra of a braided vector space of diagonal type We begin by the formal deﬁnition of PBW-basis. 6. Let A be an algebra, P, S ⊂ A and h : S → N ∪ {∞}. Let also < be a linear order on S. Let us denote by B(P, S, <, h) the set p se11 . . set t : t ∈ N0 , s1 > · · · > s t , si ∈ S, 0 < ei < h(si ), p∈P . On Nichols Algebras with Generic Braiding 51 If B(P, S, <, h) is a basis of A, then we say that (P, S, <, h) is a set of PBW generators with height h, and that B(P, S, <, h) is a PBW-basis of A.