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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
I have added to star product some basic facts, and their proofs, for the case of star products induced from constant rank-2 tensors $\omega$ on Euclidean spaces: the definition, proof of the associativity, proof that shifts of $\omega$ by symmetric contributions are algebra isomorphisms.
added statement and proof of the integral representation of the star product: here
I have spelled out the proof that the Moyal star product of a symplectic vector space is the convolution algebra of the polarized sections on the corresponding symplectic groupoid (hence is the “2-geometric quantization”): here.
This is the statement first claimed by Weinstein 91, then spelled out by Garcia-Bondia & Varilly 94, section 5. I get less dizzy with my version of the proof, but that’s just me.
I’m tempted to think of the symplectic pair groupoid as the action groupoid of the vector space on itself. This would require a small change in the symplectic structure to remain isomorphic to what you have, but it might be interesting to see if this viewpoint leads to generalisation.
More precisely it should be thought of as the action of the dual vector space on the symplectic vector space, where a covector is regarded as a linear Hamiltonian and acts via flow along its (constant) Hamiltonian vector field, which is given by contracting it with the Poisson tensor.
Aha, even better!
I have fixed a bunch of signs and prefactors in the (completely elementary) proof that the symmetric part of a star product may be shifted, up to isomorphism: here
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