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## Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• 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 $\omega$ on Euclidean spaces: the definition, proof of the associativity, proof that shifts of $\omega$ 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