On axioms of Frobenius like structure in the theory of arrangements
On axioms of Frobenius like structure in the theory of arrangements
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论排列理论中的类弗罗贝尼乌斯结构公理
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
A. Varchenko
中科院分区：
文献类型：
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作者：
A. Varchenko
Frobenius流形是一个流形，在流形各点的切空间上具有平面度规和Frobenius代数结构，使得乘法的结构常数由流形上的势函数在平面坐标下的三阶导数给出。
A Frobenius manifold is a manifold with a flat metric and a Frobenius algebra structure on tangent spaces at points of the manifold such that the structure constants of multiplication are given by third derivatives of a potential function on the manifold with respect to flat coordinates.
In this paper we present a modification of that notion coming from the theory of arrangements of hyperplanes. Namely, given natural numbers $n>k$, we have a flat $n$-dimensional manifold and a vector space $V$ with a nondegenerate symmetric bilinear form and an algebra structure on $V$, depending on points of the manifold, such that the structure constants of multiplication are given by $2k+1$-st derivatives of a potential function on the manifold with respect to flat coordinates. We call such a structure a {it Frobenius like structure}. Such a structure arises when one has a family of arrangements of $n$ affine hyperplanes in $C^k$ depending on parameters so that the hyperplanes move parallely to themselves when the parameters change. In that case a Frobenius like structure arises on the base $C^n$ of the family.