Homotopic morphisms between curved A-infinity algebras

2012-01-27T12:47:14Z

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?

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.

Answer by Leonid Positselski for Homotopic morphisms between curved A-infinity algebras

2012-01-28T21:09:49Z

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.

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).

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 $m_1=m_2=m_3=\dotsb=0$.

Similarly, any curved $A_\infty$-module over a (nonunital or strictly unital) curved $A_\infty$-algebra with a nonzero curvature element is contractible.

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, http://arxiv.org/abs/0905.2621, Remark 7.3.

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.

Answer by eigenbunny for Homotopic morphisms between curved A-infinity algebras

2012-01-28T22:10:24Z

Fukaya-Oh-Ohta-Ono's two volumes have a complete discussion of this (from memory, don't have the volumes with me now).

Homotopic morphisms between curved A-infinity algebras - MathOverflow

Homotopic morphisms between curved A-infinity algebras

Answer by Leonid Positselski for Homotopic morphisms between curved A-infinity algebras

Answer by eigenbunny for Homotopic morphisms between curved A-infinity algebras