Invariant theory of a class of infinite-dimensional groups.

*(English)*Zbl 1043.22009The authors study the theory of unitary representations of infinite-dimensional groups and their Lie algebras. One such class is the class of tame representations of inductive limits of classical groups as originated by the work of G. I. Ol’shanskiĭ [Funct. Anal. Appl. 22, No. 4, 273–285 (1988; Zbl 0668.22008)]. As in Weyl’s case of the classical groups, the authors discovered a new type of invariants in the study of concrete realizations of irreducible tame representations of inductive limits of classical groups [see T.-T. Tuong, J. Phys. A 32, 5975–5991 (1999; Zbl 0939.22010) and Helv. Phys. Acta 72, No. 4, 221–249 (1999; Zbl 0957.22014)]. One type of invariants is the Casimir invariants. Several generalizations to the case of infinite-dimensional groups may be found in [loc. cit.] and the references quoted there.

In the paper under review the authors investigate in a systematic way the study of invariant theory of inductive limits of groups acting on inverse limits of modules, rings or algebras. In particular, the Fundamental Theorem of Invariant Theory is proved, the 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 and the primeness condition of the ideals of invariants are discussed. The authors’ theory is illustrated by several examples of invariants of the infinite-dimensional classical groups. In fact, in many cases the bases of the rings of invariants are not finitely generated. This led the authors to introduce the notion of inverse limit basis which naturally involves a topology on inverse limits.

In the paper under review the authors investigate in a systematic way the study of invariant theory of inductive limits of groups acting on inverse limits of modules, rings or algebras. In particular, the Fundamental Theorem of Invariant Theory is proved, the 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 and the primeness condition of the ideals of invariants are discussed. The authors’ theory is illustrated by several examples of invariants of the infinite-dimensional classical groups. In fact, in many cases the bases of the rings of invariants are not finitely generated. This led the authors to introduce the notion of inverse limit basis which naturally involves a topology on inverse limits.

Reviewer: Peter Schenzel (Halle)

##### MSC:

22E65 | Infinite-dimensional Lie groups and their Lie algebras: general properties |

13A50 | Actions of groups on commutative rings; invariant theory |

13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |