Deformation quantization of Poisson algebras
\textit{A. Lichnerowicz} [J. Differ. Geom. 12, 253-300 (1977; Zbl 0405.53024)] (and others) proposed the concept of a deformation quantization of a symplectic manifold as a formal deformation of the algebra \({\mathfrak a} = C^ \infty(M)\). The authors [Adv. Math. 85, 224-255 (1991; Zbl 0734.58011)] proved that any symplectic manifold is deformation quantizable. In the paper under review, the authors define analogously the notion of a deformation quantization of order \(k\) (cutting the space of formal series to \({\mathfrak a} \oplus \nu{\mathfrak a}\oplus\dots\oplus \nu^ k{\mathfrak a}\) for an arbitrary Poisson structure \(\{\cdot,\cdot\}\) on a manifold \(M\)). Next, they study the obstructions for a deformation quantization of order \(k\) to be extended to that of order \(k+1\). The obstruction is obtained as a de Rham-Chevalley 3- cocycle.
scientific article; zbMATH DE number 97938