Finite-dimensional pointed or copointed Hopf algebras over affine racks

Copointed Hopf algebras overS4

“…In particular, x = y and g x⊲y = g x g y g −1 x = g 2 x = g y , so x ⊲ y = y. This contradicts [GV,Lemma 3.7 (a)], which establishes that in this case g 2 x = g y . Now, assume that x ⊲ y = y.…”

“…by Lemma [GV,Lemma 5.6 (e)] and using the adjoint action of k S 4 . Since Θ is a coalgebra map we deduce c x = c for all x ∈ X, for some fixed c ∈ k. (λ y − λ g −1 yg )δ g ∈ k S 4 , y ∈ Y.…”

“…We can consider V (X, q) as a Yetter-Drinfeld module over a finite group G with a principal realization [AG3], see Example 4.1. Then, by [GV,Lemma 3.2], J 2 (q, X) = η −1 (J 2 (X, q)). Using that…”

Lifting via cocycle deformation

“…[GIV,Theorem 5.4] Set k = C and let F 5 denote the finite field of 5 elements. Consider the affine rack X = (F 5 , 2) and the constant cocycle q ≡ −1.…”

“…Keep the notation in Example 4.12. The adapted stratification of B(X, q) considered in [GIV,Theorem 5.4] is:…”