Invariant Theory of a Class of Infinite-Dimensional Groups

The representation theory of a class of infinite-dimensional groups which are inductive limits of inductive systems of linear algebraic groups leads to a new invariant theory. In this article, we develop a coherent and comprehensive invariant theory of inductive limits of groups acting on inverse limits of modules, rings, or algebras. In this context, the Fundamental Theorem of the Invariant Theory is proved, a notion of basis of the rings of invariants is introduced, and a generalization of Hilbert’s Finiteness Theorem is given. A generalization of some notions attached to the classical invariant theory such as Hilbert’s Nullstellensatz, the primeness condition of the ideals of invariants are also discussed. Many examples of invariants of the infinite-dimensional classical groups are given.