Block Systems of finite dimensional non-cosemisimple Hopf Algebras

TL;DR

This paper analyzes finite dimensional non-cosemisimple Hopf algebras by decomposing them into blocks, establishing lower bounds for their dimensions, and exploring specific cases with prime-related dimensions.

Contribution

It introduces the block system approach for non-cosemisimple Hopf algebras and derives new bounds and results for their dimensions, especially for certain prime-related cases.

Findings

Lower bounds for the dimension of non-cosemisimple Hopf algebras.

Results for dimensions 12p, 15p, 16p, 20p, and 21p where p is prime.

Structure insights via block systems.

Abstract

In this paper, we study finite dimensional non-cosemisimple Hopf algebras through its underlying coalgebra. We decompose such a Hopf algebra as a direct sum of `blocks'. Blocks are closely related to each other by `rules' and form the `block system'. Through the block system, we are able to give a lower bound for $\dim H$, where $H$ is a non-cosemisimple Hopf algebras with no nontrivial skew-primitives, and a series of results about non-cosemisimple Hopf algebras of dimension $12p, 15p, 16p, 20p\ \text{and}\ 21p$ with $p$ being a prime number.