On inner Poisson structures of a quantum cluster algebra without coefficients

TL;DR

This paper characterizes inner Poisson structures in quantum cluster algebras without coefficients, proving they are always standard and establishing their equivalence with compatible Poisson structures through the concept of locally inner Poisson structures.

Contribution

It introduces the concept of locally inner Poisson structures and proves their equivalence to locally standard and compatible Poisson structures in quantum cluster algebras without coefficients.

Findings

Inner Poisson structures are always standard in this setting.

Locally inner Poisson structures are equivalent to locally standard Poisson structures.

Locally inner Poisson structures are equivalent to compatible Poisson structures.

Abstract

The main aim of this article is to characterize inner Poisson structure on a quantum cluster algebra without coefficients. Mainly, we prove that inner Poisson structure on a quantum cluster algebra without coefficients is always a standard Poisson structure. In order to relate with compatible Poisson structure, we introduce the concept of so-called locally inner Poisson structure on a quantum cluster algebra and then show it is equivalent to locally standard Poisson structure in the case without coefficients. Based on the result from \cite{LP} we obtain finally the equivalence between locally inner Poisson structure and compatible Poisson structure in this case.