  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.
