On inner Poisson structures of a quantum cluster algebra without coefficients

Fang Li,Jie Pan

doi:10.3934/era.2021021

Abstract

<p style='text-indent:20px;'>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. 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 [<xref ref-type="bibr" rid="b7">7</xref>] we obtain finally the equivalence between locally inner Poisson structure and compatible Poisson structure in this case.</p>

Highlights

The main aim of this article is to characterize inner Poisson structure on a quantum cluster algebra without coefficients
It can be verified that μk(Σ) is a quantum seed and μk is an involution
Fq generated by all variables in X(t) is called the quantum cluster algebra Aq(Σ)

Summary

Tn connected by an edge labeled k

Fq generated by all variables in X(t) is called the quantum cluster algebra Aq(Σ) (Theorem 2.5) Let Aq be a quantum cluster algebra without coefficients, any inner Poisson structure on Aq must be a standard Poison structure. We generalize the definition to locally inner Poisson structures and find following equivalence. (Theorem 3.6) Let Aq be a quantum cluster algebra without coefficients and {−, −} a Poisson structure on Aq. The following statements are equivalent:. (3) {−, −} is compatible with Aq. The following theorem from [8] gives a correspondence between inner Poisson brackets and k-linear transformations. We study the inner Poisson structures of a quantum cluster algebra Aq with deformation matrix Λ.

And assume

Aq with kX

For any constant a and any

The last steps of the two kinds of

Laurent monomial X

Hence by Theorem