ar X iv : m at h / 05 06 44 6 v 2 [ m at h . A T ] 2 2 Ju n 20 05 STRUCTURE RELATIONS IN SPECIAL A ∞ - BIALGEBRAS RONALD

A general A∞-infinity bialgebra is a DG module (H, d) equipped with a family of structurally compatible operations ωj,i : H ⊗i → H,where i, j ≥ 1 and i+ j ≥ 3 (see [6]). In special A∞-bialgebras, ωj,i = 0 whenever i, j ≥ 2, and the remaining operationsmi = ω1,i and ∆j = ωj,1 define the underlying A∞-(co)algebra substructure. Thus special A∞-bialgebras have the form (H, d,mi,∆j)i,j≥2 subject to the appropriate structure relations involving d, the mi’s and ∆j ’s. These relations are much easier to describe than those in the general case, which require the S-U diagonal ∆P on permutahedra. Instead, the S-U diagonal ∆K on Stasheff’s associahedra K = ⊔Kn is required here (see [5]). A∞-bialgebras are fundamentally important structures in algebra and topology. In general, the homology of every A∞-bialgebra inherits an A∞-bialgebra structure [7]; in particular, this holds for the integral homology of a loop space. In fact, over a field, the A∞-bialgebra structure on the homology of a loop space specializes to the A∞-(co)algebra structures observed by Gugenheim [2] and Kadeishvili [3]. The main result of this paper is the following simple formulation of the structure relations in special A∞-bialgebras that do not involve d: Let TH denote the tensor module ofH and let e denote the top dimensional face ofKn. There is a “fraction product” on M = End (TH) (denoted here by “•”) and certain cellular cochains ξ, ζ ∈ C (K;M) such that for each i, j ≥ 2,