Linear Lie Groups All of Whose Irreducible Finite-Dimensional Not Necessarily Unitary Representations Are of Bounded Dimension and Separate the Points of the Group
A. I. Shtern
TL;DR
The paper investigates linear Lie groups in which every irreducible finite-dimensional (not necessarily unitary) representation has bounded dimension and separates points. The authors prove that such groups must be finite extensions of commutative Lie groups, employing a strategy that passes to the identity component, uses the Levi decomposition to force commutativity of the connected part, and analyzes the discrete quotient via Gel'fand–Raikov and Thoma results together with Moore's extension principles. The argument shows the connected component is commutative, the quotient is a finite extension of a commutative subgroup, and hence the whole group inherits a finite-extensions structure with a separating set of continuous characters. The results connect representation-theoretic constraints with strong structural conclusions, and extend to Hausdorff topological groups with analogous hypotheses on their finite-dimensional representations. These findings have implications for harmonic analysis and the structural theory of Lie and topological groups.
Abstract
We prove that all linear Lie groups satisfying the conditions listed in the title are finite extensions of commutative Lie groups.