Advances in Theoretical and Applied Mathematics

One of the most famous van der Waerden theorems, which was proved in 1933, claims that every finite-dimensional locally bounded representation of a semisimple compact Lie group is continuous. As was recently proved by the author, the van der Waerden theorem can be extended to finite-dimensional representations of an arbitrary semisimple Lie group. In the present paper we start the investigation of similar conditions of automatic continuity for nonsemisimple Lie groups. Our main technical result claims that every locally bounded finite-dimensional representation of a Lie group with commutative radical is continuous if the action of the semisimple part in the Levi–Maltsev decomposition has an open orbit in the radical. In particular, every finite-dimensional representation of the volume-preserving affine group SL(n, ℝ) ⋉ ℝⁿ, n ≥ 2, and of the Poincaré group is continuous if and only if this representation is locally bounded.