Appears in collection : 2023 - T2 - WS1 - GAP XVIII: Homotopy algebras and Higher Structures

Higher operations appear in the context of up to homotopy equations. For instance the pre-Lie product is a homotopy for the commutator. Such higher operations are often associated with a graphical or geometric calculus. In new work with Rivera and Wang, we find a natural Poisson double bracket as such a homotopy, which is a member of a series of even higher brackets. Interestingly the odd brackets vanish in a directed setting, but can be defined in an undirected setting. The double bracket yields a homotopy for a four term relation involving a product, a coproduct and their opposites. It dualizes to an $m_3$ multiplicaton which is part of an $A_∞$ structure with all $m_i > 4$ vanishing. Such structures were also studied in a different context by N. Iyudu, M. Kontsevich, and Y. Vlassopoulos.