Subject: a new question for the list From: John H Palmieri Date: Fri, 24 Sep 2004 13:54:27 -0700 To: Don Davis Hi Don, Here's a batch of new questions for the list. Here are some questions about E_infinity and A_infinity algebras. In addition to the operadic description of these, I've also seen a "functional" description of A_infinity algebras: an A_infinity algebra is a graded vector space A together with maps m_n : A^{\tensor n} --> A, of degree 2-n, satisfying certain properties. For example, m_1 is a map of degree 1 with (m_1)2 = 0, so (A, m_1) is a cochain complex. Question 1: Is there a similar "functional" description of E_\infinity-algebras? For example, can I view an E_infinity algebra as being an A_infinity algebra, as above, with extra conditions on the maps m_n? If A is an A_infinity algebra, then there is a result of Kadeishvili which says that there is an A_infinity algebra structure on H(A) and a quasi-isomorphism $H(A) \rightarrow A$. Question 2: Is there a similar result for E_infinity algebras? Let A be a differential graded algebra (which one can view as an A_infinity algebra with trivial higher multiplication maps). Then H(A) has an A_infinity algebra structure, and it also has Massey products. I'm expecting that these are compatible: Question 3: Suppose that a_1, ..., a_n are classes in H(A) so that the Massey product is defined. Is m_n(a_1 tensor ... tensor a_n) in ? (I'm hoping that someone can just say, "Yes, and here's a reference.") John -- J. H. Palmieri Dept of Mathematics, Box 354350 mailto:palmieri@math.washington.edu University of Washington http://www.math.washington.edu/~palmieri/ Seattle, WA 98195-4350