Subject: Compatibility in A_\infty Hopf algebras Date: Tue, 13 Feb 2001 08:40:33 -0500 From: Ron Umble As we mentioned in an earlier communication, compatibility relations among the operations in an A_\infty Hopf algebra A link the A_\infty algebra and coagebra operations via intermediate operations \omega^{i,j} : A^{\otimes i} --> A^{\otimes j}) in degree i+j-3 with i,j>1. The first and simplest such relation links the multiplication \mu and the comultiplication \psi via the intermediate operation \omega^{2,2} and relaxes the classical Hopf condition up to homotopy, i.e., d(\omega^{2,2}) = (\mu \otimes \mu)(2,3)(\psi \otimes \psi) - \psi \mu. Now if A is a (strict) dg Hopf algebra, H(A) (thought of as a dgha with d=0) is endowed with a A_\infty Hopf algebra structure whenever \omega^{2,2} is non-zero. This phenomenon can be observed, for example, in the rational (co)homology of the double loops on a 2-connected X. For some applications (e.g., the double cobar construction on C_*(X) for any X), these relations can be defined in terms of a diagonal on the associahedra {K_n}. But in general, the relations must be defined in terms of a diagonal on the permutahedra {P_n}. The general approach is necessary, for example, in settings where all higher order A_\infty (co)algebra operations vanish but \omega^{i,j} \neq 0, i,j>1. The rational (co)homology mentioned above has this property. The {P_n} are in some sense "free", whereas the {K_n} have "torsion." Nevertheless, the two diagonals are compatible in the sense that the diagonal on P_n descends to the one on K_{n+1} under Tonk's cellural projection P_n --> K_{n+1}. This suggests that the appropriate setting for studying A_\infty Hopf compatibility might be from within Bordman and Vogt's theory of PROP's rather than from within the theory of operads. Finally, it is worth noting that the homology of every dg Hopf algebra, which need be neither free nor cofree, is endowed with an A_\infty Hopf algebra structure. Thus, A_\infty Hopf algebras not only exist, but are ubiquitous and stand among the fundamental structures in algebraic topology. Samson Saneblidze sane@rmi.acnet.ge Ron Umble ron.umble@millersville.edu