A functional ansatz is developed which gives certain elliptic solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equation. This is based on the elliptic trilogarithm function introduced by Beilinson and Levin. For this to be a solution results in a number of purely algebraic conditions on the set of vectors that appear in the ansatz, this providing an elliptic version of the idea, introduced by Veselov, of a V-system.
Rational and trigonometric limits are studied together with examples of elliptic V-systems based on various Weyl groups. Jacobi group orbit spaces are studied: these carry the structure of a Frobenius manifold. The corresponding almost dual structure is shown, in the A_N and B_N and conjecturally for an arbitrary Weyl group, to correspond to the elliptic solutions of the WDVV equations.
Transformation properties, under the Jacobi group, of the elliptic trilogarithm are derived together with various functional identities which generalize the classical Frobenius-Stickelburger relations.
A solution to the quantum Yang--Baxter equation associated with a universal two-valued algebraic group
Steklov Mathematical Institute of the Russian Academy of Sciences（俄罗斯科学院斯捷克洛夫数学研究所）；Steklov International Mathematical Center（斯捷克洛夫国际数学中心）；Centre of Integrable Systems, P.G. Demidov Yaroslavl State University（雅罗斯拉夫尔国立大学普·格·杰米多夫可积系统中心）
Resonance Relations for Solutions of the Elliptic QKZB Equations, Fusion Rules, and Eigenvectors of Transfer Matrices of Restricted Interaction-round-a-face Models
