Quantisation ideals, canonical parametrisations of the unipotent group and consistent integrable systems
M. A. Chirkov, A. V. Mikhailov, D. V. Talalaev
TL;DR
The paper develops quantisation ideals to produce quantum solutions of the Zamolodchikov tetrahedron equation in a unipotent-group setting, extending the framework to discrete dynamics and connecting it to Lusztig parametrisations and cluster algebras. It provides a classification of stable PBW ideals for $N(3,A)$ and a partial classification for $N(4,A)$, showing all such deformations are toric in origin and typically possess a classical limit, with some cases yielding non-commutative quantum algebras without a commutative limit. In the classical limit, the authors obtain a pencil of compatible Poisson brackets on the unipotent group coordinates, preserved by mutations, and identify Casimir functions and integrable pairs of invariants. These results establish a bridge between quantum reductions of noncommutative reparametrisations and mutation-invariant Poisson geometry, with potential extensions to Lusztig varieties and cluster-manifold dynamics.
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.