Home Algebra • Algebra and number theory. An integrated approach by Dixon M., Kurdachenko L., Subbotin I.

Algebra and number theory. An integrated approach by Dixon M., Kurdachenko L., Subbotin I.

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.

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