Homogeneous star products on duals of Lie algebroids are classified by second Lie algebroid cohomology, and projectable reductions commute with this classification.
Convergence of Star Products: From Examples to a General Framework
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Classification and Reduction of Homogeneous Star Products
Homogeneous star products on duals of Lie algebroids are classified by second Lie algebroid cohomology, and projectable reductions commute with this classification.