Weinstein's conjecture on tangential star products for Lie algebra duals
Let be a Lie algebra with corresponding Lie group . A tangential star product on the dual is a star product compatible with the symplectic-leaf decomposition into coadjoint orbits. The cotangent bundle has a decomposition into the left (or right) translates of coadjoint orbits.
Weinstein's conjecture. The dual admits a tangential star product if and only if there is a flat torsion-free affine connection on a neighborhood of the identity in for which the induced connection on the vector bundle is compatible with the decomposition into the left (or right) translates of coadjoint orbits.
This conjecture relates tangential deformation quantization of the linear Poisson structure on a Lie algebra dual to affine-geometric structure on the corresponding Lie group. The supplied text does not state whether the conjecture is resolved.
References
Primary source
Joshua Lackman, “A Formal Equivalence of Deformation Quantization and Geometric Quantization (of Higher Groupoids) and Non-Perturbative Sigma Models”, arXiv:2303.05494 (2023).
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