On the Green rings of pointed, coserial Hopf algebras

The Green ring is a powerful mathematical tool used to codify the interactions between representations of groups and algebras. This ring is spanned by isomorphism classes of finite-dimensional indecomposable representations that are added together via direct sums and multiplied via tensor products.

In this thesis, we explore the Green rings of a class of Hopf algebras that form an extension of the Taft algebras. These Hopf algebras are pointed and coserial, meaning their simple comodules are 1-dimensional, and their comodules possess unique composition series respectively. The comodules of these Hopf algebras thus have a particularly well-behaved structure.

We present results giving structure to the comodule Green ring of the Hopf algebra H_s and in particular fully classify the Green rings of H_s where s ≤ 6. More generally, we classify the indecomposable comodules of H_s and their composition series and prove how the composition series may be used to classify the tensor product of indecomposable comodules.

Additionally, for these Hopf algebras we classify the Grothendieck rings, the subrings of the corresponding Green rings consisting only of isomorphism classes of projective indecomposable comodules. We describe a simpler presentation of these Grothendieck rings and the multiplication in the ring.