Date: Tue, 6 Feb 2001 17:25:44 -0600 (CST) From: Ezra Getzler Subject: Another response on Hopf algebras A Hopf algebra H over an operad O is an algebra H over the operad O in the category, not of modules, but of coalgebras. In order for this to make sense, the operad O has to be an operad in the category of coalgebras, in other words, a Hopf operad. Now, the A_infinity operad is not a Hopf operad. (This is easily checked by hand: there is no possible definition of the comultiplication compatible with the operad composition.) However, the A_infinity operad is simply a cofibrant resolution of the associative operad in the closed model category of dg operads, and the associative operad IS a Hopf operad. Paul Goerss and I have proved that the category of dg Hopf operads is a closed model category, and hence there is a cofibrant resolution of the associative operad here. The Hopf algebras over this algebra give a possible definition of "Hopf algebra up to homotopy". Note that our construction is rather general, and was motivated by Tamarkin's earlier solution of the problem of finding a cofibrant resolution of the associative operad in the closed model category of dg Hopf operads (math.QA/9803025). On the other hand, there is no way to define a symmetric monoidal structure on the category of A_infinity-coalgebras: thus, it is not possible to define operads in this category, and certainly not to make any sense of compatibility conditions for A_infinity products and coproducts. -- Ezra Getzler http://athos.math.nwu.edu Department of Mathematics, (847) 467-1695 (work) Northwestern University, (773) 868-4947 (home) Evanston, IL 60208-2730, USA (847) 491-8906 (fax)