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

Using the method of quantisation ideals, we construct a family of quantisations corresponding to Case α 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 coeficients. 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). Our decomposition of the unipotent group is explicitly connected to that introduced by Lusztig, which makes links with the theory of cluster algebras. However 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.