Root multiplicities for Nichols algebras of diagonal type of rank two

Abstract

Abstract. We determine the multiplicities of a class of roots for Nichols algebras of diagonal type of rank two, and identify the corresponding root vectors. Our analysis is based on a precise description of the relations of the Nichols algebra in the corresponding degrees.

On some root multiplicities for Nichols algebras of diagonal type over arbitrary fields

“…By applying the similar discussions in the proof of [4,Lemma 4.11] for m = n − j n − 1, k = j n , we conclude that H = q m 12 q −(n−jn)(n+jn−1)/2 n j n q (−r) −m Q jn,n−jn−1…”

“…Let us fix the total ordering < on X such that x i < x j if and only if 1 ≤ i < j ≤ n. Let X and X × denote the set of words and non-empty words with letters in X, respectively. We write < lex for the lexicographic ordering which is induced by the total ordering < on X (for detail see [4,Section 3]). For any word w = x i 1 x i 2 · · · x is ∈ X, we write |w| = s for the length of w.…”

“…We end this part by recalling some results and some important polynomials in T (V ) from [4]. Lemma 3.9] Let m ∈ N 0 .…”

“…Remark 3.20. If the p = 0, then Theorem 3.19 is exactly Theorem 4.16 in [4]. It can happen that J 2 = ∅, even if p is prime number.…”

“…The elements(ad x 1 ) m−j (P j ) for j ∈ J 1 are linearly independent in ker(π) ∩ U m .Proof. It was proved in[4, Theorem 4.16].…”