Exceptional sequences and Drinfeld double Hall algebras

TL;DR

This paper studies the structure of Drinfeld double Hall algebras associated with hereditary categories, showing invariance under mutations and that they are generated by any complete exceptional sequence in key cases.

Contribution

It provides explicit mutation formulas in Drinfeld double Hall algebras and proves invariance of subalgebras generated by exceptional sequences under mutation equivalences.

Findings

Subalgebras generated by exceptional sequences are mutation-invariant.

Double composition algebra is generated by any complete exceptional sequence in specific categories.

Parallel results are established for the Lie algebra case.

Abstract

Let $\A$ be a finitary hereditary abelian category and $D(\A)$ be its reduced Drinfeld double Hall algebra. By giving explicit formulas in $D(\A)$ for left and right mutations, we show that the subalgebras of $D(\A)$ generated by exceptional sequences are invariant under mutation equivalences. As an application, we obtain that if $\A$ is the category of finite dimensional modules over a finite dimensional hereditary algebra, or the category of coherent sheaves on a weighted projective line, the double composition algebra of $\A$ is generated by any complete exceptional sequence. Moreover, for the Lie algebra case, we also have paralleled results.