Bialgebra structure on Bridgeland's Hall algebra of two-periodic complexes

TL;DR

This paper investigates the bialgebra structure of Bridgeland's Hall algebra of two-periodic complexes, establishing a coproduct and showing its equivalence to the Drinfeld double in the hereditary case.

Contribution

It introduces a coproduct on Bridgeland's Hall algebra and proves its bialgebra structure matches the Drinfeld double for hereditary categories.

Findings

Coproduct defined on Bridgeland's Hall algebra.

Bialgebra structure coincides with Drinfeld double in hereditary case.

Provides new insights into the algebraic structure of two-periodic complexes.

Abstract

We study the bialgebra structure of the Hall algebra of two-periodic complexes recently introduced by Bridgeland. We introduce coproduct on Bridgeland's Hall algebra, and show that in the hereditary case the resulting bialgebra structure coincides with that on Drinfeld double of the ordinary Hall algebra