
Journal of Lie Theory 13 (2003), No. 2, 401425 Copyright Heldermann Verlag 2003 Invariant Theory of a Class of InfiniteDimensional Groups Tuong TonThat Dept. of Mathematics, University of Iowa, Iowa City, IA 52242, U.S.A. ThaiDuong Tran Dept. of Computer Science, Southwest Texas University, 601 University Drive, San Marcos, TX 78666, U.S.A. The representation theory of a class of infinitedimensional 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 infinitedimensional classical groups are given. Keywords: Invariant theory, inductive limits, groups acting on inverse limits of modules, rings, algebras, Fundamental Theorem of Invariant Theory. MSC: 13A50; 22E65, 13F20 [ Fulltextpdf (296 KB)] for subscribers only. 