论文标题
半不变的symétriquesde收缩抛物面
Semi-invariants symétriques de contractions paraboliques
论文作者
论文摘要
令$ k $为具有特征零的代数封闭字段，而$ \ mathfrak {g} $ a lie代数。令$ y（\ mathfrak {g}）$为对称代数$ s（\ mathfrak {g}）= k [\ mathfrak {g}^*] $由多项式制成的对称代数$ s（\ mathfrak {g}）= k [\ mathfrak {g}）= k [\ mathfrak {g}^*] $。还将$ sy（\ mathfrak {g}）$定义为由$ s（\ mathfrak {g}）$的元素生成的代数，其伴随动作在同一个上行动。当$ \ mathfrak {q} $是类型$ a $ a $或$ c $的抛物线收缩时，在某些情况下，在$ b $的某些情况下，Panyushev和Yakimova表明，不变的代数$ y（\ Mathfrak {q}）$是polynomials的代数。使用Panyushev和Yakimova的结果，我们通过在类型$ a $中构造一组代数免费的发电机，在某些情况下以$ c $中的类型来显示$ sy（\ mathfrak {q}）$的多项式。我们还研究了$ c $的示例，其中$ sy（\ mathfrak {q}）$不是多项式。
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.
