%0 Journal Article
%T $A_{\infty}$-structures on an elliptic curve
%A Alexander Polishchuk
%J Mathematics
%D 2000
%I arXiv
%R 10.1007/s00220-004-1078-7
%X The main result of this paper is the proof of the "transversal part" of the homological mirror symmetry conjecture for an elliptic curve which states an equivalence of two $A_{\infty}$-structures on the category of vector bundles on an elliptic curves. The proof is based on the study of $A_{\infty}$-structures on the category of line bundles over an elliptic curve satisfying some natural restrictions (in particular, $m_1$ should be zero, $m_2$ should coincide with the usual composition). The key observation is that such a structure is uniquely determined up to homotopy by certain triple products.
%U http://arxiv.org/abs/math/0001048v2