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\({{\varvec{A}}}_{\varvec{\infty }}\)-algebras associated with elliptic curves and Eisenstein–Kronecker series

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We compute the \(A_{\infty }\)-structure on the self-\({\text {Ext}}\) algebra of the vector bundle G over an elliptic curve of the form \(G=\bigoplus _{i=1}^r P_i\oplus \bigoplus _{j=1}^s L_j\), where \((P_i)\) and \((L_j)\) are line bundles of degrees 0 and 1, respectively. The answer is given in terms of Eisenstein–Kronecker numbers \((e^*_{a,b}(z,w))\). The \(A_\infty \)-constraints lead to quadratic polynomial identities between these numbers, allowing to express them in terms of few ones. Another byproduct of the calculation is the new representation for \(e^*_{a,b}(z,w)\) by rapidly converging series.

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Correspondence to Alexander Polishchuk.

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Supported in part by the NSF Grant DMS-1400390 and by the Russian Academic Excellence Project ‘5-100’.

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Polishchuk, A. \({{\varvec{A}}}_{\varvec{\infty }}\)-algebras associated with elliptic curves and Eisenstein–Kronecker series. Sel. Math. New Ser. 24, 563–589 (2018).

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