Affiliation:
1. Institute of Mathematics, Fraser Noble Building, University of Aberdeen, Aberdeen AB24 3UE, UK
Abstract
Abstract
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 $\textrm{GL} (W)$ 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 $K[U(W)]^{\textrm{GL} (W)}$. 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 $K[U(W)]^{\textrm{GL} (W)}$. 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 $K[U(W)]^{\textrm{GL} (W)}$ can be calculated explicitly from $K[X]$ in this case.
Publisher
Oxford University Press (OUP)
Cited by
1 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献