In this paper, all simple Yetter-Drinfeld modules and indecomposable projective Yetter-Drinfeld modules over the $2$-rank Taft algebra $\overline{\mathcal{A}}$ are construted and classified by Radford's method of constructing Yetter-Drinfeld modules over a Hopf algebra. Furthermore, the projective class ring of the category of Yetter-Drinfeld modules over $\overline{\mathcal{A}}$ is described explicitly by generators and relations.

Let ${\mathrm{\Delta }}_q$ be the universal Askey-Wilson algebra. If $q$ is not a root of unity, it is shown in the Huang's earlier paper that an $(n+1)$-dimensional irreducible ${\mathrm{\Delta }}_q$-module is a quotient $V_n(a,b,c)$ of a ${\mathrm{\Delta }}_q$-Verma module with

$$ConditionA:abc,a^{-1}bc,a{b}^{-1}c,ab{c}^{-1}\notin \left\{q^{n-2i+1}|1\le i\le n\right\}.$$

The aim of this paper is to discuss the structures of $(n+1)$-dimensional ${\mathrm{\Delta }}_q$-modules $V_n(a,b,c)$ when the given triples $(a,b,c)$ do not satisfy Condition A.

In this paper, all simple Yetter-Drinfeld modules and indecomposable projective Yetter-Drinfeld modules over a family of non-pointed $8m$-dimension Hopf algebras of tame type with rank two, are construted and classified. The technique is Radford's method of constructing Yetter-Drinfeld modules over a Hopf algebra. Furthermore, the projective class rings of the category of Yetter-Drinfeld modules over this class of Hopf algebras are described explicitly by generators and relations.

The aim of this paper is to characterize the representation ring of a non-pointed and noncocommutative bialgebra. First, the isomorphism classes of its indecomposable modules are classified. Then the tensor product of modules is established. Finally, its representation ring is described.

In this paper, we focus on studying two classes of finite dimensional $\mathrm{\Delta }$-associative algebras, which are extensions of a family of $2n^2$-dimensional Kac-Paljutkin type semi-simple Hopf algebras $H_{2n^2}$. All their indecomposable modules are classified. Furthermore, their representation rings are described by generators with suitable relations.