二元三次代数的特殊辛结构
The special symplectic structure of binary cubics
摘要：
设$k$是特征不为2或3的域。设$V$是二元三次多项式的$k$ -空间. $k^2$上的自然辛结构提升为$V$上的辛结构$ω$，并且从$\textrm{Sl}(2,k)$的自然辛作用得到辛模$(V,ω)$。我们给出了一个完整的分析，这个辛模的角度来看，相关的时刻地图，其规范平方$Q$（基本上是经典的判别式）和辛梯度的$Q$。其中的结果是一个辛推导的Cardano-Tartaglia公式的根的一个三次，详细的参数为所有$\textrm{Sl}(2,k)$和$\textrm{Gl}(2,k)$ -轨道，特别是确定一组结构上的一组$\textrm{Sl}(2,k)$ -轨道的固定非零判别式，和一个纯粹的辛推广经典爱森斯坦syzygy的协变的二进制三次。这种精细的辛分析是由于特殊的辛性质继承了周围例外李代数$\mathfrak G_2$。
英文摘要：
Let $k$ be a field of characteristic not 2 or 3. Let $V$ be the $k$-space of binary cubic polynomials. The natural symplectic structure on $k^2$ promotes to a symplectic structure $ω$ on $V$ and from the natural symplectic action of $\textrm{Sl}(2,k)$ one obtains the symplectic module $(V,ω)$. We give a complete analysis of this symplectic module from the point of view of the associated moment map, its norm square $Q$ (essentially the classical discriminant) and the symplectic gradient of $Q$. Among the results are a symplectic derivation of the Cardano-Tartaglia formulas for the roots of a cubic, detailed parameters for all $\textrm{Sl}(2,k)$ and $\textrm{Gl}(2,k)$-orbits, in particular identifying a group structure on the set of $\textrm{Sl}(2,k)$-orbits of fixed nonzero discriminant, and a purely symplectic generalization of the classical Eisenstein syzygy for the covariants of a binary cubic. Such fine symplectic analysis is due to the special symplectic nature inherited from the ambient exceptional Lie algebra $\mathfrak G_2$.
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