1997/09/30 by Alan Weinstein, Ping Xu, Weinstein, Alan +1
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Quantum Algebra (math.QA) #dg-ga #math.DG #math.QA #q-alg
paper · pdf · doi:10.48550/arxiv.q-alg/9709043
LaTex2e, 22 pages, minor corrections in the last section and a reference added (Festschrift for V.I. Arnol'd's 60th birthday, V2, AMS, Providence, to appear)
arxiv created 1997/10/05 · arxiv updated 2009/11/30
We show that the Hochschild cohomology of the algebra obtained by formal deformation quantization on a symplectic manifold is isomorphic to the formal series with coefficients in the de Rham cohomology of the manifold. The cohomology class obtained by differentiating the star-product with respect to the deformation parameter is seen to be closely related to the characteristic class of the quantization. A fundamental role in the analysis is played by ``quantum Liouville operators,'' which rescale the deformation parameter in the same way in which Liouville vector fields scale the Poisson structure (or the units of action). Several examples are given.