arXiv 2007.038452020-07-08math.RTmath.QA

泛不变量环

Universal rings of invariants

Ehud Meir

备注：: 38 pages, 13 figures

摘要：

设$K$是特征为零的代数闭域。给定的$K$上的向量空间$W$上的特定类型的代数结构（例如代数或余代数）可以被编码为仿射空间$U(W)$中的点。这个空间具有一个$\text{GL}(W)$作用，两个点定义同构结构当且仅当它们位于同一轨道上。这导致研究不变量环$K[U(W)]^{\text{GL}(W)}$。我们用生成元和关系来描述这个环。然后，我们组合地构造一个交换环$K[X]$，它专门针对$K[U(W)]^{\text{GL}(W)}$形式的所有不变量的环。我们证明了交换环$K[X]$具有更丰富的Hopf代数结构，它具有额外的余积、分次和内积，从而使它成为一个有理的PSH-代数，推广了Zelevinsky引入的结构.我们完成了详细的研究$K[X]$的情况下，由一个单一的自同态代数结构，并显示如何环的不变量$K[U(W)]^{\text{GL}(W)}$可以计算明确从$K[X]$在这种情况下。

英文摘要：

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.

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。

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