Invariants for algebraic group actions

研究环的不变量有一个漫长而辉煌的历史（见，例如，希尔伯特的第十四问题）。其中一个亮点的理论在二十世纪是证明环的不变量约化代数群作用于（坐标环）仿射品种是numerous产生-这意味着有一个有用的概念商品种在这种情况下。如果群作用不是约化的，有限生成一般是失败的，但通常可以恢复理论的某些方面。其中一个例子是代数群作用的“分离不变量”的概念--这些不变量是有限的不变量集，可以分离整个不变量环，即使整个不变量环不是连续生成的。该项目的目的是研究非还原情况下代数群作用的不变量，首先建立在Dufresne和合著者的工作基础上，为加法群作用的不变量。

The study of rings of invariants has a long and illustrious history (see, for example, Hilbert's fourteenth problem). One of the highlights of the theory in the twentieth century was the proof that rings of invariants for reductive algebraic groups acting on (the coordinate rings of) affine varieties are finitely generated -- this implies that there is a useful notion of quotient variety in this situation. Finite generation fails in general if the group acting is not reductive, but it is often possible to retrieve some aspects of the theory. One example of this is the notion of "separating invariants" for algebraic group actions -- these are finite sets of invariants which can separate as much as the whole ring of invariants, even when the full ring of invariants is not finitely generated. The aim of this project is to study invariants for algebraic group actions in non-reductive cases, starting by building on work of Dufresne and co-authors for invariants of additive group actions.