Subject: Re: three postings From: jds@sas.upenn.edu Date: Mon, 27 Sep 2004 10:02:40 -0400 Response to Palmieri's #3 Yes, and the chapter on Massey products in LNM 161 H-spaces froim a homtopy point of view nibbles around the full generality you want jim >> >> Subject: a new question for the list >> From: John H Palmieri >> Date: Fri, 24 Sep 2004 13:54:27 -0700 >> >> 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 >> ___________________________________________________________________ Subject: Re: three postings From: "A Lazarev, Mathematics" Date: Mon, 27 Sep 2004 15:03:40 +0100 Here's a partial answer to John's questions. 1)Concerning a functional description of E_\infty algebras. In general, there isn't one and I suspect that even if there was, it would be extremely complicated. However, if your ground ring k contains the rational numbers, then there is a convenient definition of an E-infinity algebra in which case it is usually called C-infinity. Suppose that k is a field of char 0 for simplicity. Let V be a graded k-vector space and consider A=\hat{L}(\Sigma V*). Here * stands for the k-dual, \Sigma is the shift functor, L is the free Lie algebra and \hat stands for completion. In other words, you are looking at formal Lie series in \Sigma V^*. Then a C-infinity structure is a continuous derivation m of A of degree 1 having square 0 and without constant term. If you write down the condition m2=0 in coordinates you get a complicated system of relations which resemble the definition of an A-infinity algebra. Note that an A-infinity algebra can be defined in exactly the same way, replacing L with T, the tensor algebra. And yes, you can view a C-infinity algebra as an instance of an A-infinity algebra since a derivation of a free Lie algebra is also a derivation of the corresponding tensor algebra. 2)Yes, there is a minimality theorem for C-infinity algebras which states that any C-infinity algebra is weakly equivalent to a minimal one. A minimal C-infinity algebra is such that the derivation m has no linear term. This looks like a different definition of minimality than the one you meant, but it is in fact the same, only phrased differently. I don't know about the E-infinity case and strongly doubt that such a theorem would hold. 3)Here I don't have anything constructive to offer. I am fairly certain that such a result is not found in the so-called mathematical literature. However, there is a huge body of physical papers which deals with infinity algebras, sometimes calling them that, sometimes not. No one can be sure what is proved and what is not in those papers :-) Andrey. ___________________________________________________________________ Subject: Re: three postings From: "Brooke E. Shipley" Date: Mon, 27 Sep 2004 10:10:19 -0500 An answer to John Palmieri's third question: See: Leif Johansson and Larry Lambe, Transferring Algebra Structures Up to Homology Equivalence, Scand. Math., 89 (2001), no. 2, 181--200. Also available at: http://www.math.su.se/~lambe/public/pubs.html 6.1 gives a connection between the A_infinity "functional" structure and triple Massey products. This can be extended to all triples by considering matric Massey products. I couldn't find a reference for n-tuples when I was looking for one in 2001. Brooke Shipley ______________________________________________________________ Subject: Re: three postings From: Ezra Getzler Date: Mon, 27 Sep 2004 13:31:55 -0500 (CDT) Here are some answers to John Palmieri's questions. These questions only have good answers in characteristic zero. In that case, we may replace E_infinity-algebras by what John Jones and I called C_infinity algebras in our paper on E_k-operads; namely, instead of only assuming that the products m_n are Hochschild cochains, one assumes that they are Harrison cochains - that is, they vanish on shuffles. (This is an elegant generalization of the condition that m_2 is commutative.) I am not sure about Kadeishvili's proof of the result on going down from an A_infinity-algebra A to its cohomology H(A), but Merkulov gave a proof with explicit formulas for the operations on H(A). If in fact A is a C_infinity algebra, then the same formulas yield not only an A_infinity structure on H(A), but a C_infinity structure. This has been proved in a recent thesis of my student Xuezhi CHeng at Northwestern University. (In his book, Smirnov states this result, but without proof. Since this book should never have been published by the AMS, it being so full of mistakes, I am not sure that it is safe use this citation in a paper, although here, he did not make a mistake.) Ezra Getzler