# Drinfeld doubles via derived Hall algebras and Bridgeland Hall algebras

## TL;DR

This paper presents a new algebraic framework connecting Drinfeld doubles with derived and Bridgeland Hall algebras, providing explicit realizations and extending understanding of these structures in the context of hereditary categories.

## Contribution

It offers a Hall algebra presentation of Kashaev's theorem and new realizations of Drinfeld double Hall algebras via derived and Bridgeland Hall algebras.

## Key findings

- Hall algebra presentation of Kashaev's theorem
- Realizations of Drinfeld double Hall algebra via derived Hall algebra
- Realizations via Bridgeland Hall algebra of m-cyclic complexes

## Abstract

Let $\A$ be a finitary hereditary abelian category. We give a Hall algebra presentation of Kashaev's theorem on the relation between Drinfeld double and Heisenberg double. As applications, we obtain realizations of the Drinfeld double Hall algebra of $\A$ via its derived Hall algebra and Bridgeland Hall algebra of $m$-cyclic complexes.