Primitive elements in Ringel-Hall algebras of tame hereditary algebras

TL;DR

This paper characterizes primitive elements in the Ringel-Hall algebra of tame hereditary algebras, generalizing previous results and providing explicit bases for these elements.

Contribution

It offers a comprehensive description of primitive elements in H(A) for tame hereditary algebras, extending and refining prior work on tame quivers.

Findings

Explicit description of primitive elements in H(A)

An identity for primitive elements in the subalgebra generated by regular modules

Construction of an explicit basis for primitive elements

Abstract

We study primitive elements in the Ringel-Hall algebra H(A) of an algebra A over a finite field associated with a quiver with automorphism. When A is a tame hereditary algebra, we give a description of primitive elements in H(A) which generalizes and improves a result of Hennecart (IMRN 2021) for tame quivers. Moreover, we obtain an identity concerning primitive elements in the subalgebra of H(A) generated by regular A-modules which enables us to construct an explicit basis for the space of primitive elements in H(A).