Rational solutions of the Witten–Dijkgraaf–Verlinde–Verlinde (or WDVV) equations of associativity are given in terms of configurations of vectors, which satisfy certain algebraic conditions known as ⋁-conditions [A. P. Veselov, Phys. Lett. A 261, 297–302 (1999)]. The simplest examples of such configurations are the root systems of finite Coxeter groups. In this paper, conditions are derived that ensure that an extended configuration—a configuration in a space one-dimension higher—satisfies these ⋁-conditions. Such a construction utilizes the notion of a small orbit, as defined in Serganova [Commun. Algebra, 24, 4281–4299 (1996)]. Symmetries of such resulting solutions to the WDVV equations are studied, in particular, Legendre transformations. It is shown that these Legendre transformations map extended-rational solutions to trigonometric solutions, and for certain values of the free data, one obtains a transformation from extended ⋁-systems to the trigonometric almost-dual solutions corresponding to the classical extended affine Weyl groups.

中文翻译：

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WDVV方程的扩展⋁系统和三角解

Witten-Dijkgraaf-Verlinde-Verlinde（或WDVV）关联性方程的有理解是根据向量的配置给出的，这些向量满足某些称为⋁条件的代数条件[AP Veselov，Phys。来吧 阿261，297-302（1999）]。此类配置的最简单示例是有限的Coxeter组的根系统。在本文中，推导了确保扩展配置（在空间上高一维的配置）满足这些⋁条件的条件。这种结构利用了Serganova [Commun。代数，24，4281–4299（1996）]。研究了WDVV方程的此类结果解的对称性，尤其是Legendre变换。结果表明，这些勒让德变换将扩展的有理解映射到三角解，并且对于自由数据的某些值，人们获得了从扩展的system系统到对应于经典扩展仿射Weyl基的三角几乎对偶解的变换。