Quantisation ideals, canonical parametrisations of the unipotent group and consistent integrable systems

TL;DR

This paper develops a new family of quantisations for unipotent matrices using quantisation ideals, linking to cluster algebras and integrable systems, and analyzing their classical limits and Poisson structures.

Contribution

It introduces a novel family of quantisations related to Sergeev's solutions of the tetrahedron equation, with new Poisson brackets and integrable systems compatible with mutations.

Findings

Constructed quantisations corresponding to Sergeev's classification

Developed a family of compatible Poisson brackets invariant under mutations

Identified integrable systems compatible with parametrisation mutations

Abstract

Using the methods of quantisation ideals, we construct a family of quantisations corresponding to Case alpha in Sergeev's classification of solutions to the tetrahedron equation. This solution describes transformations between special parametrisations of the space of unipotent matrices with noncommutative coefficients. We analyse the classical limit of this family and construct a pencil of compatible Poisson brackets that remain invariant under the re-parametrisation maps (mutations). This decomposition problem is closely related to Lusztig's framework, which makes links with the theory of cluster algebras. Our construction differs from the standard family of Poisson structures in cluster theory; it provides deformations of log-canonical brackets. Additionally, we identify a family of integrable systems defined on the parametrisation charts, compatible with mutations.