# Bridgeland's Hall algebras and Heisenberg doubles

## TL;DR

This paper introduces the $m$-periodic lattice algebra for a hereditary algebra and proves its isomorphism with Bridgeland's Hall algebra, also embedding the Heisenberg double Hall algebra into it, revealing new algebraic structures.

## Contribution

It defines the $m$-periodic lattice algebra and establishes its isomorphism with Bridgeland's Hall algebra, also embedding the Heisenberg double Hall algebra, advancing understanding of algebraic structures related to hereditary algebras.

## Key findings

- The algebra $L_m(A)$ is isomorphic to Bridgeland's Hall algebra $ ext{DH}_m(A)$.
- An embedding of the Heisenberg double Hall algebra into $ ext{DH}_m(A)$ is constructed.
- The paper connects lattice algebras with Hall algebras in the context of hereditary algebras.

## Abstract

Let $A$ be a finite dimensional hereditary algebra over a finite field, and let $m$ be a fixed integer such that $m=0$ or $m>2$. In the present paper, we first define an algebra $L_m(A)$ associated to $A$, called the $m$-periodic lattice algebra of $A$, and then prove that it is isomorphic to Bridgeland's Hall algebra $\mathcal {D}\mathcal {H}_m(A)$ of $m$-cyclic complexes over projective $A$-modules. Moreover, we show that there is an embedding of the Heisenberg double Hall algebra of $A$ into $\mathcal {D}\mathcal {H}_m(A)$.