Hall algebras as Poisson algebras

Abstract

Motivated by Joyces work on motivic Hall algebras, we observe that for an algebra $\Lambda$ with Hall polynomials,the Hall algebra $\mathcal{H}(\Lambda)$ of $\Lambda$ admits a natural structure of Poisson algebra. For a given antisymmetric bilinear form over its Grothendieck group $\go(\Lambda)$satisfying certain condition, there is a homomorphism of Poisson algebras from the Hall algebra $\mathcal{H}(\Lambda)$ to its associated torus algebra. In particular, the condition holds true forrepresentation-finite cluster-tilted algebras.