Invariant theory of a class of infinite-dimensional groups

Tuong Ton-That and Thai-Duong Tran

Abstract: The representation theory of a class of infinite-di\-men\-sion\-al 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 {\it Fundamental Theorem of the Invariant Theory} is proved, a notion of {\it basis} of the rings of invariants is introduced, and a generalization of {\it Hilbert's Finiteness Theorem} is given. A generalization of some notions attached to the classical invariant theory such as {\it 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.

Keywords: Invariant theory of inductive limits of groups acting on inverse limits of modules, rings, or algebras, Fundamental Theorem of Invariant Theory

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