Homotopic morphisms between curved A-infinity algebras - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T01:11:44Z http://mathoverflow.net/feeds/question/86821 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/86821/homotopic-morphisms-between-curved-a-infinity-algebras Homotopic morphisms between curved A-infinity algebras Ed Segal 2012-01-27T12:47:14Z 2012-01-28T22:10:24Z <p>I know how to think about (curved) $A_\infty$-algebras 'geometrically', i.e. via formal non-commutative geometry in the sense of Kontsevich etc. I also know how to think about $A_\infty$-morphisms in this way. But what if two $A_\infty$-morphisms are homotopic? Does anyone know how to interpret this fact geometrically?</p> <p>This is particularly important in the curved situation, because then there's no such thing as a quasi-isomorphism, so we only have homotopy-equivalence. Also I have a vague memory of reading something about this (probably written by Kontsevich), but I've searched all the papers I can think of and not found it.</p> http://mathoverflow.net/questions/86821/homotopic-morphisms-between-curved-a-infinity-algebras/86920#86920 Answer by Leonid Positselski for Homotopic morphisms between curved A-infinity algebras Leonid Positselski 2012-01-28T21:09:49Z 2012-01-28T21:09:49Z <p>I don't know specifically about homotopies, but the notion of a curved $A_\infty$-algebra is generally problematic. In the conventional setting of algebras over a field, it is just trivial in the following strong sense.</p> <p>Let $A$ and $B$ be two curved $A_\infty$-algebras over a field $k$ with nonzero curvature elements $m_{0,A}\ne0\ne m_{0,B}$. This is a suffient condition to trivialize nonunital curved $A_\infty$-algebras; in the (strictly) unital case, assume that $m_{0,A}$ and $m_{0,B}$ do not belong to the one-dimensional vector subspaces generated by the units of $A$ and $B$ (whcich could happen in the $\mathbb Z/2$-graded case).</p> <p>Then any isomorphism of graded vector spaces $f\colon A\to B$ (preserving the units, in the unital case) can be extended to an $A_\infty$-isomorphism $(f_0,f_1,f_2,\dotsc)\colon A\to B$ with $f_0=0$ and $f_1=f$. So there precisely as many curved $A_\infty$-algebras with nonzero curvature, up to $A_\infty$-isomorphism, as there are graded vector spaces; and any curved $A_\infty$-algebra with a nonzero curvature is $A_\infty$-isomorphic to a curved $A_\infty$-algebra with $m_1=m_2=m_3=\dotsb=0$.</p> <p>Similarly, any curved $A_\infty$-module over a (nonunital or strictly unital) curved $A_\infty$-algebra with a nonzero curvature element is contractible.</p> <p>These results are mostly due to Kontsevich; I learned them from conversations with him while visiting IHES and subsequently recorded them in what is now AMS Memoir vol.212 #996, 2011, <a href="http://arxiv.org/abs/0905.2621" rel="nofollow">http://arxiv.org/abs/0905.2621</a>, Remark 7.3.</p> <p>It appears that if you want to have a nontrivial theory of curved $A_\infty$-algebras, you have to do it over, say, a local ring and require the curvature elements in your algebras to be divisible by the maximal ideal of the local ring. I am presently working on this; the writeup is available from my homepage.</p> http://mathoverflow.net/questions/86821/homotopic-morphisms-between-curved-a-infinity-algebras/86927#86927 Answer by eigenbunny for Homotopic morphisms between curved A-infinity algebras eigenbunny 2012-01-28T22:10:24Z 2012-01-28T22:10:24Z <p>Fukaya-Oh-Ohta-Ono's two volumes have a complete discussion of this (from memory, don't have the volumes with me now).</p>