Déformations d'algèbres associées à une variété symplectique (les $*_\nu $-produits)

Remarks on Symplectic Connections

“…This was first observed by Lichnerowicz [16] for a restricted class of star products. Theorem 1.5 [12] A natural star product at order 2 determines a unique symplectic connection.…”

Closedness of Star Products and Cohomologies

“…[23,24]) that it is always possible to avoid the obstructions; and if two star products are equivalent to order k, once they are made to coincide at that order the skew-symmetric part of their difference at order k +1 determines a closed 2-form that is exact iff they are equivalent to order k + 1. (This follows from an argument due to S. Gutt, similar to those of [19,22]). …”

“…where T (2) is a differential operator of order at most 2,b denotes the Chevalley coboundary operator and Λ 2 is a 2-tensor, image (under µ −1 ) of a closed 2-form [19,22]. A somewhat general expression can also be given for C 4 [19], but it is much more complicated.…”

“…In particular when the C r are differentiable and satisfy the parity condition, the C 2r+1 being n.c. (what is called a "weak star product"), any star product is equivalent [22] to a "Vey star product", one for which the r!C r have the same principal symbol as P r Γ , the r th power of the Poisson bracket expressed with covariant derivatives ∇ relative to some symplectic connection Γ, i.e. on a local chart U (with summation convention on repeated indices):…”

Symplectic Connections

“…Fix four points a, b, c, d. Regarding Definition 8.3 and formula (33), one needs to construct our volume preserving diffeomorphism ϕ : (M, µ) → (M, µ) in such a way that for all t,…”