A B∞-algebra is an algebraic structure encoding higher multibraces and homotopy associativity, first encountered by Baues in relation to iterated loop spaces. More famous is the example of a B∞ structure on the Hochschild cochain complex: the relation of B∞ to G∞ algebras was crucial for some proofs of Deligne's Hochschild cohomology conjecture.
We present work in progress (joint with M Ronco and A Tonks) investigating the appearance of canonical B∞-algebra structures on any algebra endowed with an independent differential graded algebra structure.
A special case of a commutative analogue of this is the appearance of Koszul's L∞ brace hierarchy, of importance in the BRST approach to closed string theory. Other special cases we rely on are due to Börjeson, Markl and Loday-Ronco.