Representations of a Class of Drinfeld's Doubles

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.…”

“…If M is of (2, 2)-type and contains a submodule of (1, 1)-type, then M is isomorphic to the module M 2 (1, r, η), where r ∈ Z 2 , η = ∞ or η ∈ k. Using the pullback diagrams given in [8, p. 15], one can describe the structures of the modules M 2 (1, r, η), r ∈ Z 2 , η = ∞ or η ∈ k. If M is of (2, 2)-type and does not contain any submodule of (1, 1)-type, then M is isomorphic to a submodule of 2P (1, r) for some r ∈ Z 2 (see [8]). Using the structure of P (1, r), one can prove that M has the structure described in (11) below (see Lemma 2.30).…”

“…In order to study the left D 4 -module algebras of dimension 4, we need first to give the structures of all indecomposable D 4 -modules with dimension not exceeding 4. We will follow the notations of [8].…”

Generic Modules Over a Class of Drinfeld's Quantum Doubles

“…From the results given in Chen (2005), the basic algebra of any nonsimple block of H n 1 q is independent of n and q. Therefore, it suffices to describe the infinite representations of H 2 1 −1 .…”

“…In the previous articles (Chen, 1999(Chen, , 2000(Chen, , 2002(Chen, , 2005, 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 , the integers, H p q has an n 4 -dimensional quotient Hopf algebra H n p q .…”