论文标题
不变的通用环
Universal rings of invariants
论文作者
论文摘要
令$ k $为特征零的代数封闭字段。特定类型的代数结构（例如，在$ k $上的给予矢量空间$ w $ w $ w $ the $ k $的代数结构（例如代数或山地）可以用作仿射空间$ u（w）$中的点。该空间配备了$ \ text {gl}（w）$ action，并且两个点在且仅当它们位于同一轨道中时定义了同构结构。这导致研究不变的环$ k [u（w）]^{\ text {gl}（w）} $。我们通过生成器和关系描述了这枚戒指。然后，我们在组合上构造了一个交换环$ k [x] $，该$专门用于表格$ k [u（w）]^{\ text {gl}（w）} $的所有不变的环。我们表明，交换环$ k [x] $具有更丰富的HOPF代数结构，并具有额外的coproduct，评分和内部产品，使其成为理性的PSH-Elgebra，从而推广了Zelevinsky引入的结构。在代数结构的情况下，我们完成了$ k [x] $的详细研究，该代数结构由单个内态组成，并展示如何从$ k [x] $中明确计算不变的$ k [u（w）]^{\ text {gl}（w）} $。
Let $K$ be an algebraically closed field of characteristic zero. Algebraic structures of a specific type (e.g. algebras or coalgebras) on a given vector space $W$ over $K$ can be encoded as points in an affine space $U(W)$. This space is equipped with a $\text{GL}(W)$ action, and two points define isomorphic structures if and only if they lie in the same orbit. This leads to study the ring of invariants $K[U(W)]^{\text{GL}(W)}$. We describe this ring by generators and relations. We then construct combinatorially a commutative ring $K[X]$ which specializes to all rings of invariants of the form $K[U(W)]^{\text{GL}(W)}$. We show that the commutative ring $K[X]$ has a richer structure of a Hopf algebra with additional coproduct, grading, and an inner product which makes it into a rational PSH-algebra, generalizing a structure introduced by Zelevinsky. We finish with a detailed study of $K[X]$ in the case of an algebraic structure consisting of a single endomorphism, and show how the rings of invariants $K[U(W)]^{\text{GL}(W)}$ can be calculated explicitly from $K[X]$ in this case.
