The Stable Picard Group of A(n)

TL;DR

This paper determines the structure of the stable Picard group of the algebra A(n) for n≥2, showing it is isomorphic to Z⊕Z by analyzing endotrivial modules and using reductions to subalgebras.

Contribution

It provides a complete description of the stable Picard group of A(n) for n≥2, a result not previously established, using novel reduction techniques.

Findings

Stable Picard group of A(n) is Z⊕Z for n≥2

Endotrivial modules are key to the analysis

Reductions from Hopf algebra to subalgebras are effective

Abstract

In this paper, we showed that the Stable Picard group of $A(n)$ for $n\geq 2$ is $\mathbb{Z}\oplus \mathbb{Z}$ by considering the endotrivial modules over $A(n)$. The proof relies on reductions from a Hopf algebra to its proper Hopf subalgebras.