抛物线压缩的半不变量对称性
Semi-invariants symétriques de contractions paraboliques
摘要：
设$K$是特征为零的代数闭域，$\mathfrak{g}$是李代数.设$Y(\mathfrak{g})$是由在伴随作用下不变的多项式构成的对称代数$S(\mathfrak{g})=K[\mathfrak{g}^*]$的子代数。也定义$Sy(\mathfrak{g})$为伴随作用同位作用的$S(\mathfrak{g})$的元素生成的代数。当$\mathfrak{q}$是一个抛物线收缩型$A$或$C$，并在某些情况下，在类型$B$，Panyushev和Yakimova表明，代数的不变量$Y(\mathfrak{q})$是一个代数的多项式。利用Panyushev和Yakimova的结果，通过构造$A$型和某些情况下$C$型生成元的代数自由集，证明了$Sy(\mathfrak{q})$的多项式性.我们还研究了$C$类型中的一个例子，其中$Sy(\mathfrak{q})$不是多项式。
英文摘要：
Let $K$ be an algebraically closed field with characteristic zero, and $\mathfrak{g}$ a Lie algebra. Let $Y(\mathfrak{g})$ be the subalgebra of the symmetric algebra $S(\mathfrak{g})=K[\mathfrak{g}^*]$ made of the polynomials which are invariant under the adjoint action. Also define $Sy(\mathfrak{g})$ as the algebra generated by elements of $S(\mathfrak{g})$ for which the adjoint action acts homothetically. When $\mathfrak{q}$ is a parabolic contraction in type $A$ or $C$, and in some cases in type $B$, Panyushev and Yakimova showed that the algebra of invariants $Y(\mathfrak{q})$ is an algebra of polynomials. Using Panyushev's and Yakimova's result, we show the polynomiality of $Sy(\mathfrak{q})$ by constructing an algebraically free set of generators in type $A$ and in some cases in type $C$. We also study an example in type $C$ where $Sy(\mathfrak{q})$ is not polynomial.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
