Intrinsic Brackets and the L∞-deformation Theory of Bialgebras

We show that there exists a Lie a bracket on the cohomology of any type of (bi)algebras over an operad or a prop, induced by an L∞-structure on the defining cochain complex, such that the associated L∞-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 [27] provide explicit formulas for this bracket. Conventions. We assume a certain familiarity with operads and props, see [22, 18, 23, 24, 30]. The reader who wishes only to know how the intrinsic bracket on the GerstenhaberSchack cohomology looks might proceed directly to Section 6 which is almost independent on the rest of the paper and contains explicit calculations. We also assume some knowledge of the concept of strongly homotopy Lie algebras (also called L∞-algebras), see [12, 15]. We will make no difference between an operad P and the prop P generated by this operad. This means that for us operads are particular cases of props. As usual, bialgebra will mean a Hopf algebra without (co)unit and antipode. To distinguish these bialgebras from other types of “bialgebras” we will sometimes call them also Ass-bialgebras. All algebraic objects will be defined over a fixed field k of characteristic zero although, surprisingly, our constructions related to Ass-bialgebras make sense over the integers.