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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 28th 2017

    I have added to star product some basic facts, and their proofs, for the case of star products induced from constant rank-2 tensors ω\omega on Euclidean spaces: the definition, proof of the associativity, proof that shifts of ω\omega by symmetric contributions are algebra isomorphisms.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 19th 2017

    added statement and proof of the integral representation of the star product: here

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 20th 2017
    • (edited Dec 20th 2017)

    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.

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeDec 20th 2017
    • (edited Dec 20th 2017)

    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.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeDec 21st 2017

    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.

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeDec 21st 2017

    Aha, even better!

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJan 3rd 2018

    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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