Universal rings of invariants
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 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)]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)]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)]GL(W) can be calculated explicitly from K[X] in this case.