We show that there exists a Lie bracket on the cohomology of anytype of (bi)algebras over an operad or a \PROP, induced by an $L_\infty$-structure on the defining cochain complex, such that the associated $L_\infty$-master equation captures deformations.
This in particular implies the existence of a Lie bracket on the Gerstenhaber-Schack cohomology [7] of a bialgebra that extends the classical intrinsic bracket [6] on the Hochschild cohomology, giving an affirmative answer to an old question about the existence of such a~bracket. We also explain how the results of [24] provide explicit formulas for this bracket.
Journal of Homotopy and Related Structures, Vol. 5(2010), No. 1, pp. 177-212