  Associated to a differential BV algebra are two differential graded Lie
algebras: we call one classical and the other, which contains a formal h-bar
parameter, quantum. The classical dgLa is always smooth formal. In this paper,
we give necessary and sufficient conditions for the quantum dgLa to be smooth
formal. These conditions are equivalent to the degeneration of a version of the
noncommutative Hodge to de Rham spectral sequence. References added.
