A class of noncommutative and noncocommutative hopf algebras: the quantum version

Finite-Dimensional Representations of a Quantum Double

“…In particular, we have H n p q ∼ = H n 1 q ∼ = D A n q −1 for any p = 0. For details, the reader can see [3,4]. Thus in order to discuss the representations of D A n q −1 , we only have to consider the representations of H n 1 q .…”

“…In the previous papers [3,4], we constructed an infinite-dimensional noncommutative and noncocommutative Hopf algebra H p q for any p q ∈ k with q = 0, where k is a fixed field. When q is a root of the nth cyclotomic polynomial over Z, H p q has an n 4 -dimensional quotient Hopf algebra H n p q .…”

“…We also recall the structures of A n ω and H n p q . The material of this section appears in [1][2][3][4][5].…”

Four-dimensional Yetter–Drinfeld module algebras over H4

“…The Drinfeld doubles of Taft's Hopf algebras and their finite representations were investigated in [5][6][7][8]. Let us recall some results which we need throughout the paper.…”

“…Therefore, we may assume 1 ∈ 2V (1, 0) as the identity 1 generates a copy of V (1, 0). Thus we may choose a linear basis {x 1 , x 2 , x 3 , 1} for A such that {x 1 , x 2 } is a canonical basis for M 1 (1, 0, ∞) with the D 4 -action given by (5) and 2V (1, 0) 4 . Replacing x 3 with x 3 + α for some suitable α ∈ k, we may assume…”