Semi-derived Ringel-Hall bialgebras

TL;DR

This paper establishes a bialgebra structure on the semi-derived Ringel-Hall algebra of a hereditary abelian category, showing its compatibility with the Drinfeld double Ringel-Hall algebra structure.

Contribution

It introduces a coproduct formula for the semi-derived Ringel-Hall algebra and proves its compatibility, endowing it with a bialgebra structure.

Findings

The coproduct on $SH( )$ is explicitly formulated.

The coproduct is compatible with the algebra product.

The bialgebra structure matches that of the Drinfeld double Ringel-Hall algebra.

Abstract

Let $\mathcal{A}$ be an arbitrary hereditary abelian category. Lu and Peng defined the semi-derived Ringel-Hall algebra $SH(\mathcal{A})$ of $\mathcal{A}$ and proved that $SH(\mathcal{A})$ has a natural basis and is isomorphic to the Drinfeld double Ringel-Hall algebra of $\mathcal{A}$. In this paper, we introduce a coproduct formula on $SH(\mathcal{A})$ with respect to the basis of $SH(\mathcal{A})$ and prove that this coproduct is compatible with the product of $SH(\mathcal{A})$, thereby the semi-derived Ringel-Hall algebra of $\mathcal{A}$ is endowed with a bialgebra structure which is identified with the bialgebra structure of the Drinfeld double Ringel-Hall algebra of $\mathcal{A}$.