Triangular dynamical r-matrices and quantization

TL;DR

This paper explores the geometric and algebraic structures of triangular dynamical r-matrices, demonstrating their connection to Poisson manifolds and establishing a classification of their quantizations using Lie algebra cohomology.

Contribution

It introduces a Poisson geometric framework for triangular dynamical r-matrices and classifies their quantizations via relative Lie algebra cohomology, extending to splittable cases.

Findings

Triangular dynamical r-matrices induce regular Poisson manifolds.

Non-degenerate cases are quantizable and classified by cohomology.

Generalization to splittable cases broadens applicability.

Abstract

We provide a general study for triangular dynamical r-matrices using Poisson geometry. We show that a triangular dynamical r-matrix always gives rise to a regular Poisson manifold. Using the Fedosov method, we prove that non-degenerate (i.e., the corresponding Poisson manifolds are symplectic) triangular dynamical r-matrices (over $ \frakh^* $ and valued in $\wedge^{2}\frakg$) are quantizable, and the quantization is classified by the relative Lie algebra cohomology $H^{2}(\frakg, \frakh)[[\hbar ]]$. We also generalize this quantization method to splittable triangular dynamical r-matrices, which include all the known examples of triangular dynamical r-matrices. Finally, we arrive a conjecture that the quantization for an arbitrary triangular dynamical r-matrix is classified by the formal neighbourhood of this r-matrix in the modular space of triangular dynamical r-matrices. The dynamical r-matrix cohomology is introduced as a tool to understand such a modular space.