转置不可逆多项式
Transposing Noninvertible Polynomials
摘要：
Landau-Ginzburg镜像对称预言了分次Frobenius代数（记作$\mathcal{A}$和$\mathcal{B}$）之间的同构，这些分次Frobenius代数是由非退化拟齐次多项式$W$和相关的对称群$G$构造的。对于$W$和$G$的特定选择，证明了$\mathcal{A}$和$\mathcal{B}$模型之间的对偶性。在$W$被限制为具有与变量相同数量的单项式（称为$\textit{invertible}$）的许多情况下，这些结构已经得到了证明。当允许$W$有比变量多的单项式时，对$\mathcal{A}$和$\mathcal{B}$模型之间的同构作了一些说明.在本文中，我们证明这些命题是假的，也就是说，不存在的命题同构。洞察这个问题不仅会产生新的结果朗道-金兹伯格镜像对称，但也将是有趣的，从一个纯粹的代数的角度来看，作为一个结果，对集团的作用在分次代数。
英文摘要：
Landau-Ginzburg mirror symmetry predicts isomorphisms between graded Frobenius algebras (denoted $\mathcal{A}$ and $\mathcal{B}$) that are constructed from a nondegenerate quasihomogeneous polynomial $W$ and a related group of symmetries $G$. Duality between $\mathcal{A}$ and $\mathcal{B}$ models has been conjectured for particular choices of $W$ and $G$. These conjectures have been proven in many instances where $W$ is restricted to having the same number of monomials as variables (called $\textit{invertible}$). Some conjectures have been made regarding isomorphisms between $\mathcal{A}$ and $\mathcal{B}$ models when $W$ is allowed to have more monomials than variables. In this paper we show these conjectures are false; that is, the conjectured isomorphisms do not exist. Insight into this problem will not only generate new results for Landau-Ginzburg mirror symmetry, but will also be interesting from a purely algebraic standpoint as a result about groups acting on graded algebras.
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