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キーワード:
Mathematics, Algebraic Geometry, Nonlinear Sciences, Exactly Solvable and Integrable Systems
要旨:
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.
