From: Ron Umble Subject: RE: 5 answers and a question Date: Tue, 6 Feb 2001 11:51:10 -0500 _______________________________ In response to: Subject: question for list: A_infinity Hopf algebras From: palmieri@math.washington.edu (John H. Palmieri) Date: 05 Feb 2001 13:58:43 -0800 A colleague of mine wants to know about A_infinity Hopf algebras. Does anyone know a definition, or a reference for a definition? I would guess that it's both an A_infinity algebra and an A_infinity coalgebra, compatibly, but I'm not sure what "compatibly" should mean. ______________________________________ John, The background is in our preprint AT/0011065 in the archive, which defines a diagonal on the associahedra and a tensor product in the A_\infty category. Beyond that, one needs to generalize the notion of an (f,g)-derivation homotopy to what we call a \Delta-derivation homotopy with respect to a compatible family of (higher) homotopies. As you say, an A_\infty Hopf algebra is simultaneously an A_\infty algebra and an A_\infty coalgebra, but there are "intermediate" operations \omega^{i,j} : A^{\otimes i} --> A^{\otimes j} in degree i+j-3 that extend to \Delta-derivation homotopies wrt a certain family of compositions of maps in lower degrees. Intuitively, think of \omega^{i,j} as the i+j-3 cell in K_{i+j-1}. The relevant family consists of compositions of \omega^{i,j}'s that make up the boundary. In reality, the \omega^{i,j}'s induce maps of tilde cobar constructions on iterated tensor products of A_\infty coalgebras; these are the \Delta derivation homotopies referred to above. A preprint should be out sometime this year; in the meantime the best we can do is share our lecture notes if that would help. Ron Umble cc: Samson Saneblidze