论文标题
通勤方程及其三角解决方案
Commutativity equations and their trigonometric solutions
论文作者
论文摘要
我们考虑通勤方程$ f_i f_j = f_j f_i $用于函数$ f（x^1，\ dots，x^n），其中$ f_i $是三阶衍生物$ f_ {ikl} $的矩阵。我们表明，在某些非分类条件下，解决方案$ f $满足WDVV方程。同等地，弗罗贝尼乌斯代数的相应家族具有身份字段$ e $。我们还研究三角解决方案$ f $由有限的矢量收集确定的，我们为所有已知的解决方案提供了明确的公式。向量的相应集合通过非简单的根系系统给出，或与它们对镜子交点的投影有关。
We consider commutativity equations $F_i F_j =F_j F_i$ for a function $F(x^1, \dots, x^N),$ where $F_i$ is a matrix of the third order derivatives $F_{ikl}$. We show that under certain non-degeneracy conditions a solution $F$ satisfies the WDVV equations. Equivalently, the corresponding family of Frobenius algebras has the identity field $e$. We also study trigonometric solutions $F$ determined by a finite collection of vectors with multiplicities, and we give an explicit formula for $e$ for all the known such solutions. The corresponding collections of vectors are given by non-simply laced root systems or are related to their projections to the intersection of mirrors.
